Reply to Deleted user Good OP, Lionino. Switching logic into natural languages was a big handicap in my last thread. It seemed to be a simple riddle for everyone, until I asked to explain it with natural language and whether the concepts of ambiguity and contradictory are similar or not. I only got answers using logic language constantly until RussellA wrote a very good example using natural language.
I have to agree that statements like A?B are universal, and I guess it helps people use logic quickly and easily. But, again, it is outstanding to see those logic formulas explained in language. It seems they are only allowed to use it with "A" and "B" in the premises.
Eventually, you will even need to add quantifiers (? ?) and predicates to express in logic something as simple as:
[i]All humans are mortal.
Socrates is human.
Therefore Socrates is mortal.[/i]
If you want to express in logic statements about logic itself -- which is a requirement for philosophical statements -- you even need to add support for arithmetic.
The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic.
Reply to Tarskian let me rephrase: it doesn't match MY intuition, and many other people. To many of us, (2+2=4) implies (Kamala Harris is a presidential nominee) makes no sense even if the classical logic truth table comes out as true, because the left side of the implication at least seemingly has nothing to do with the right side.
Maybe it matches your intuition, and I'm sorry for trying to speak for you. My mistake.
let me rephrase: it doesn't match MY intuition, and many other people.
It is probably a mixup between the implication, which is just a truth table, and the entailment, a ? b, which means that consequent b necessarily follows from antecedent a.
(2+2=4) ? (Kamala Harris is a presidential nominee)
is false, because the consequent cannot be justified from the antecedent.
So, it is rather about a mixup in vocabulary than about intuition. I guess that many other people do that indeed too.
As I alluded to in the other thread, material implication captures English usage only insofar as it guarantees that if the antecedent is true then the consequent will also be true. Similarly, the negation of a material implication says that if the antecedent is true then the consequent will be false, and this is vaguely similar to the denial of an implication in English except for the fact that the falsity of the consequent is not guaranteed in English.
The key is that in English we prescind from many things that material implication does not prescind from, such as the value of the consequent in that denial case. As another example, if an antecedent is false then the material implication is true, whereas this does not hold in English. At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not.
I think in examining these we are combining two confusing and non-translatable logical concepts: material implication and contradiction. Neither one translates well into English, and their combination translates especially badly.
Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).
The key is that in English we prescind from many things that material implication does not prescind from
For example, one can assert the material implication (P?Q) for three reasons:
P is true and Q is true
P is false (and Q is true)
P is false (and Q is false)
In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this.
In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values.
Exactly.
It represents an entailment A ? B, and not just a simple implication A?B.
Logic makes all its decisions by only looking at truth values while the English version assumes the existence of a system that also investigates justification.
If A does not imply B and [regardless of whether] B is false, can we really infer that A is true?
Yes, because it means A without B. Isn't it intuitive that A without B entails A? And isn't it intuitive that A?B means not A without B, i.e. ¬(A ? ¬B), so that ¬(A?B) means A without B, and therefore A ? ¬B and therefore A?
Logic isn't a replacement for natural languages. Nor is it a set of rules for how one ought construct arguments. This was part of the subject of my thread Logical Nihilism, and the work of Gillian Russell.
in English, there is no lexical distinction between inclusive-or and exclusive-or, but A?B is inclusive-or, meaning the result is also True if both are True.
So what logic does in this case is to set out explicitly two ways of using "or" of which we were probably unaware. After understanding this we are able to say clearly whether we are using an exclusive or an inclusive "or". Prior to that logical analysis, we were probably unaware of the distinction, let alone which we were using.
So logic here is setting up a degree of precision that can carry over into natural languages. It's acting as a tool to make clear what it is we are doing with our sentences.
It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances. Rather logic sets out sub-games within language, with their own specific rules. Natural languages permit the breaking of the rules of any of these sub-games.
Take a look at these examples from Russell. ? ? ? and ? & ? ? ? might seem to be candidates for logical laws one might expect to have complete generality.
Identity: ? therefore ?;: a statement implies itself. But consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"
Elimination: ? and ? implies ?; But consider "? is true only if it is part of a conjunction".
Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule.
Logic doesn't give us a crystalline replacement for natural languages. But it can set out clearly what it is we are doing with our statements.
"A does not imply B". In English that is ambiguous. It could mean:
There are instances in which A is true but B is false.
It is not the case that A entails B (same as above).
It is not the case that A implies B (where 'implies' means the material conditional).
It is not the case that A implies B (where 'implies' means a connective other than the material conditional).
Probably others.
The rest of this pertains to ordinary symbolic logic:
We have to be careful to distinguish between, on the one hand, mere implication and, on the other hand, and entailment or proof .
A -> B
is not generally equivalent with
A |= B or A |- B.
In ordinary symbolic logic, '->' does not mean 'entails' or 'proves':
A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).
A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.
A |- B iff and only if there is a derivation of B from A.
Example:
"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.
"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.
"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln".
/
Also, we need to be careful what we mean by letters such as 'A', 'B', 'P', 'Q', etc.
(In propositional logic, all formulas are sentences, but in predicate logic, some formulas are sentences and some formulas are not sentences.)
In different contexts, such letters are used to represent either:
(1) atomic formulas (atomic sentences)
or
(2) meta-variables ranging over formulas. (Sometimes logic books use Greek letters for this.)
In recent discussions, the letters are being used as meta-variables.
So, for example, when we mention 'A -> B', we understand that 'A' and 'B' range over all sentences, including ones of arbitrary complexity.
/
If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:
"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs
The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional.
[EDIT: Dump the strikethrough potion]
[s]Arguably, they are the equivalent:
(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional)
is equivalent with:
(2) "If A then B" if and only if "There are no instances in which A is true and B is false"
So:
(4) "It is not the case that every instance in which A is true is an instance in which B is true"
is equivalent with
(3) "It is not the case that there are no instances in which A is true and B is false"
is equivalent with:
(5) "There are instances in which A is true and B is false"[/s]
First, that is not idiomatic. I've never heard someone say "There is X without there is Y". Second, it could mean at least a few different things. Third, I don't know your point with the example. Fourth, the previous example at least had a nice haiku-like quality.
Saying A?B is "if A then B" does not provide a solution to the matter of unambiguously converting A?B to English.
'A -> B' is symbolic. In context of ordinary symbolic logic, it is unambiguous. What is ambiguous is everyday discourse. And, of course, many ordinary senses of "if then" don't fit 'A -> B' as 'A -> B' is used in ordinary symbolic logic. What you call an 'incongruity' stems from (1) "If then" has different sense in ordinary discourse. (2) The material conditional is not in accord with many (arguably, most) everyday senses of "if then".
The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this.
"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower.
Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).
In the example I gave, "First-order claims and second-order rules of discourse."
It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances.
...
Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule.
Yet if what Aristotle does in Metaphysics IV is correct, then there is a logical law that cannot be breached, namely the law of non-contradiction. Or in other words, "logic" is not a purely formal exercise. It was created for a reason and that reason has implications for reality/metaphysics.
But for the natural speaker what is lacking is a relation between the two things.
The natural speaker assumes that there is somewhere some justification.
Formal languages may expect that too.
That is actually the main difference between classical logic and mathematical logic.
In mathematical logic, it is not just about truth tables. The goal is not limited to a bit of truth value calculus. The goal is proving entailment, i.e. (mathematical) justification.
Reply to Banno Because you said in the post I quoted above: "It's acting as a tool to make clear what it is we are doing with our sentences."
And, that's what syntax is about. The arrangement of words and phrases in a specific order to make clear what we are doing with our sentences. Transposing them could change the meaning. So, syntax is the specific tool to make our phrases clear or let's say, 'understandable'. I don't attempt to deny the value of logic in all of this. I simply think that this is a subject of linguistics rather than logic.
Reply to Deleted user If syntax is not a tool for working with sentences, what is the main point of syntax then?
Does logic make clear what we do with sentencing as Banno suggested?
I can’t see how ‘P(a)?Q(a)’ helps me to properly write: ‘the cute dog ate the bone’ for example.
Ah, I see the problem, and I carelessly extended it.
I'm dumping this:
"If A then B" if and only if "Every instance in which A is true is an instance in which B is true".
That is wrong.
As I mentioned before, there are two different notions:
(1) "If A then B"
and
(2) "A entails B"
(1) in the sense of material implication means "(A is true and B is true) or (A is false and B is true) or (A is false and B is false)". And that reduces to "A is false or B is true".
(2) means "Every instance in which A is true is an instance in which B is true".
Since Aristotle, the assumption that consistency is a requirement for truth, validity, meaning, and rationality, has gone largely unchallenged. Modern investigations into dialetheism, in pressing the possibility of inconsistent theories that are nevertheless meaningful, valid, rational, and true, call that assumption into question. If consistency does turn out to be a necessary condition for any of these notions, dialetheism prompts us to articulate why; just by pushing philosophers to find arguments for what previously were undisputed beliefs it renders a valuable service... And if consistency turns out not to be an essential requirement for all theories, then the way is open for the rational exploration of areas in philosophy and the sciences that have traditionally been closed off.
"In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q."
I speak English, and I don't take "if P then Q" (whether in the sense of material implication or in everyday senses, including necessity or relevance) to be about presences. Indeed, where 'P' and 'Q' are sentences, I would take "the sentence P is present", etc., to be nonsense unless it meant that the sentence P was being displayed in some way, such as on a page or screen. Indeed, I've never heard an English speaker in everyday conversation say something like "The sentence P is present". Moreover, let P stand for a sentence such as "The world is big", then I've never heard any English speaker say anything like ""The world is big" is present". Indeed, if an English said "If the world is big, then the sun is huge" then I don't know any English speaker who would say, "Yes, the presence of "The world is big" indicates the presence of "The sun is huge". Not only is that dialogue not idiomatic, but it registers as nonsense.
If relevance is required between the antecedent and consequent for meaningfulness, then we don't know whether a given conditional is meaningful until we've settled whether there is relevance between the antecedent and the consequent. So if the question of relevance is unsettled, we have to wait before taking the conditional to be meaningful or not. "If Jackie has blue hair then London is noiser this year than last year". We don't know whether the antecedent is relevant to the consequent without knowing more. Maybe Jackie having blue hair causes a big fashion trend in which people go to London to be seen having blue hair or many other possibilities. For that matter, when would we ever be certain that there is no relevance between two sentences? A butterfly flapping its wings in Tierra del Fuego, so to speak.
That is actually the main difference between classical logic and mathematical logic.
Usually, mathematical logic is studied by means of classical logic. Indeed, mathematical logic is formulated by classical set theory. The theorems of mathematical logic, if formalized, are themselves theorems of set theory.
Not following you here - there is more to clarity, and to logic, than just syntax.
I agree. I just wanted to point out that syntax is a tool to make clear sense of our sentences. Not the only one, for sure. But it is one of the main tools in linguistics at least.
For example: sometimes logic formulas or axioms are not clear, but thanks to syntax we can get a better approach to understand it.
Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).
Note, though, that, "You are contradicting yourself," or, "This is a contradiction," is a different genus, and deviates from first-order discourse, moving into the meta-language.
So an example of a second-order rule of discourse is, "Thou shalt not contradict thyself."
there are laws of thought that can't be broken (for obvious reasons).
What are some of those laws of thought that can't be broken but are not laws of logic? How do you state the difference between laws of logic and laws of thought? What are the obvious reasons they can't be broken?
The most problematic foundational law in logic (Boole's "laws of thought") is in my opinion the law of the excluded middle (LEM), which implicitly assumes that the question at hand is decidable.The indiscriminate use of this law is intuitionistically objectionable:
Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle. This law states that, for any proposition, either that proposition is true or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an axiom.
The law of identity may also be problematic because of the existence of indiscernible numbers. However, this problem is not frequently mentioned in the literature.
The only foundational law that seems to withstand foundational scrutiny by constructive mathematics, is the law of non-contradiction:
The law of non-contradiction (which states that contradictory statements cannot both be true at the same time) is still valid.
The law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the ground that any proof or disproof must use the law itself prior to reaching the conclusion. In other words, in order to verify or falsify the laws of logic one must resort to logic as a weapon, an act that is argued to be self-defeating.
"The laws of physics don't apply here", the meaning is clear. You yourself use the word without any apparent confusion:
for any law, there are cases in which that law does not apply
— TonesInDeepFreeze
(1) I know the ordinary general sense of 'apply'. But this is a particular subject, and I'm wondering whether you have an explication of your use or whether 'apply' should just be taken as undefined by you. (2) I was asking you about your use of 'apply'; I didn't assert my own use of it. I didn't assert what you quoted of me; it was part of a question to you.
for any law, there are cases in which that law does not apply
— TonesInDeepFreeze
This, but one can make up scenarios and/or systems where that law does not apply. That was one of the answers at least to the liar paradox: making a completely different system.
What are some of those laws of thought that can't be broken but are not laws of logic?
— TonesInDeepFreeze
I don't think there any, as soon as we can express our thoughts in language we can also express the rules our thoughts follow in language (this language being logic sometimes).
You said that there are laws of thought that can't be broken. And you said laws of logic can be broken. What are some laws of thought that can't be broken but are not laws of logic?
What are the obvious reasons they can't be broken?
— TonesInDeepFreeze
For example, I can't conceive of anything as being other than it is, because as soon as I conceive it, it is what it is, and not something else. I cannot imagine something as being otherwise.
You can't conceive it. But that doesn't entail that others cannot conceive it. Also, conceiving that a contradiction holds does not entail that the contradiction holds.
the law of the excluded middle (LEM), which implicitly assumes that the question at hand is decidable.
In context of modern logic, 'decidable' means either (1) the sentence or its negation is a theorem, or (2) There is an algorithm to decide whether the sentence is a member of a given set, such as the set of sentences that are valid, or the set of sentences that are true in a given model.
LEM is not that. LEM syntactically is the theorem: P v ~P, and LEM semantically is the theorem that for a given model M, either P is true in M or P is false in M (so, either P is true in M or ~P is true in M)
The law of identity may also be problematic because of the existence of indiscernible numbers.
The law of identity, the indiscernibility of identicals, and the identity of indiscernibles are different. What specific problem with the law of identity are you referring to?
The only foundational law that seems to withstand foundational scrutiny by constructive mathematics, is the law of non-contradiction:
You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
If — then — is only used in math/logic because it is clearer to look at than If —, —.
It's not used only in logic and mathematics. In everyday discourse, people write "If ___, then" commonly. The source you cited mentioned mentions "If ___, ___" only but I would not take that to preclude also "If ___, then". Are there grammarians who explicitly disallow it? Are there not grammarians who do allow it? Perhaps there are grammarians explicitly disallow "If ___, then ___", but that would be pedantic, especially in this context, in face of the fact that "If ___, then ___" is not only used in everyday discourse, but in all kinds of writing. Moreover, since it is taken as grammatical in logic and mathematics, then that's good enough here, since logic is the subject. I don't know what point you are making about logic when you rule out "If ___, then ___".
That is why I said "I am literally dying now" instead of "I am dying now". It is an incorrect usage of the word 'literally' if you are not really dying, therefore grammatically incorrect.
As far as I can tell, it is grammatical. 'literally' is an adjective to the noun 'dying'. But the sentence is false. "I am hopelessly dying", "I am unhappily dying", "I am literally dying". Grammatical as far as I know.
their usage of the word is often just grammatically incorrect.
What rule of grammar is violated. I wouldn't take using a word with an incorrect meaning is not a violation of grammar. If someone thought 'choleric' means 'melancholic', then "Jack is choleric" is still grammatical even though Jack is not choleric.
I would need to re-read that article, but, as I recall, dialetheism is a philosophy not a system. Though, as you mention, there are paraconsistent systems. Yes, that is an example. But, for any for any law of thought there may be a system that denies the law, so any law of thought could be denied.
If your point is that one is free to choose any system one wants to use, then, of course, one could not dispute that. But also one is free to choose whatever ways of thinking one wants to choose.
Can you conceive something as other than what it is?
Whether or not I can conceive it doesn't entail that others cannot. It is not precluded that, for example, people in mystic states do experience suspension of non-contradiction. And it does not dialetheism permit conceiving such things?
You said, "Some laws of logic may express those laws of thought. But that is just a semantic contention."
Leontiskos said laws of logic can't be broken. I said that it is the laws of thought that can't be broken instead. Despite the disagreement in choice of words, I still understand the content of his post.
I guess 'that' referred to the difference in the way you two stated the idea. Okay.
/
I asked, "Do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply?"
For example, I can't conceive of anything as being other than it is, because as soon as I conceive it, it is what it is, and not something else. I cannot imagine something as being otherwise. This reminds of the law of identity, and it just might be.
This is very close to the way that Aristotle defends the PNC in Metaphysics IV. Much of this is just a question of what we mean by 'logic'.
n context of modern logic, 'decidable' means either (1) the sentence or its negation is a theorem, or (2) There is an algorithm to decide whether the sentence is a member of a given set, such as the set of sentences that are valid, or the set of sentences that are true in a given model.
LEM is not that. LEM syntactically is the theorem: P v ~P, and LEM semantically is the theorem that for a given model M, either P is true in M or P is false in M (so, either P is true in M or ~P is true in M)
"the sentence or its negation is a theorem" ignores the existence of true but unprovable sentences. So, it should rather be "the sentence or its negation is true". They don't need to be provable theorems.
I do not see the difference between "the sentence or its negation is true" and "P v ~P".
The law of identity, the indiscernibility of identicals, and the identity of indiscernibles are different. What specific problem with the law of identity are you referring to?
I was referring to the identity of indiscernibles: ?x ?y [ ?F ( F x ? F y ) ? x = y ]
For any x and y, if x and y have all the same properties, then x is identical to y.
You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
I was referring to Boole's laws of thought:
- the law of identity (ID)
- the law of contradiction (or non-contradiction; NC)
- the law of excluded middle (EM)
https://en.wikipedia.org/wiki/Law_of_thought
The title of George Boole's 1854 treatise on logic, An Investigation on the Laws of Thought, indicates an alternate path. The laws are now incorporated into an algebraic representation of his "laws of the mind", honed over the years into modern Boolean algebra.
Boole did not "invent" these foundational laws but he did systematize them somewhat.
"the sentence or its negation is a theorem" ignores the existence of true but unprovable sentences. So, it should rather be "the sentence or its negation is true". They don't need to be provable theorems.
I'm just telling you what the definition is. It doesn't matter what you think "should" be or what "needs" to be.
You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
— TonesInDeepFreeze
I was referring to Boole's laws of thought:
- the law of identity (ID)
- the law of contradiction (or non-contradiction; NC)
- the law of excluded middle (EM)
And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
1h
Using a word to mean something other than what it does is exactly a violation of grammar.
"What it does" meaning its syntactical role, yes.
"What it means", no.
If I think 'red' means 'loud' and I say "The trombone is red", then still "The trombone is red" is grammatical even though it is false and false due to the speaker's mistake in the meaning of the word 'red'.
And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
1h
Rhetorical question: is it possible to misspeak, which is to say to speak wrongly, without committing a grammar mistake?
What does 'speak wrongly' mean? Speak ungrammatically or speak falsely?
Of course it is possible to use the wrong word and still be grammatical. People do it all the time.
One could make up examples all day, or observe them.
'literally' is an adjective. "I was literally dying" is grammatical. It is not made ungrammatical by the fact in the world that the speaker happened to not be dying and not literally dying.
My reply to Leontiskos, which you asked about, is exactly that, except that it is laws of logic that a system may deny, not laws of thought.
The way it read was that there are laws of logic that may be broken but not laws of thought. But if any law of logic may be also a law of thought, then there are laws of thought that may be broken too. And it wasn't stated as to what systems may deny, but merely as to what laws may deny.
Is grammar not the rules which give us what can be said right or wrong in language?
Grammar doesn't dictate what is true or false, only what is well formed.
"I was literally dying" is well formed even if untrue.
"Bob's French horn is red" is well formed even if untrue.
We can give millions of examples in which the speaker misuses a word, but the sentence is still grammatical. Since you are wont to skip that point, here's one more:
The speaker may think 'melancholic' means 'mellow', then say, "The song is melancholic" when the song is not at all melancholic. A false but grammatical utterance.
One may choose different ways of thinking but every way of thinking that one may choose still has fundamental rules of rationality.
What is regarded as rational may be different for different people. And people may choose even to think irrationally by any standard. But, of course, given a particular conception of rationality, some thoughts will not be rational and will violate certain attendant laws of rational thinking.
That something is necessary for rationality (under a given definition of 'rationality') doesn't entail that people may not break "laws of thought".
— TonesInDeepFreeze
I can't imagine how it does not entail unless you are working under a very thin definition of rationality.
Doesn't matter what the definition is. People may break all kinds of norms of rationality in their thinking. But, of course, tautologically, they can't break those norms with out breaking those norms.
And it does not dialetheism permit conceiving such things?
— TonesInDeepFreeze
I personally don't think dialethism is universally applicable or says anything deep about human rationality. It may be helpful as a gimmick to work around self-reference paradox, but that is about it.
Whatever one thinks about dialetheism, the point stands that people may conceive dialetheistically. A person may say of himself that he cannot conceive other than by certain rules regarded as irrational not to conceive by. But that doesn't entail that other people can't conceive outside of those rules. Indeed, in such things as art, dreams, ruminations and mystical experiences, people can conceive in all kinds of ways. But, again, if the point is that people can't think irrationally without thinking irrationally, then of course, it would be irrational to deny that point.
When people say — not lying or confused — that their cat is black, but they actually have a dog who is white, and they are thinking of their white dog but saying "My cat is black", they are using the words 'cat' and 'black' wrongly.
But not ungrammatically.
"My cat is black" is grammatical even though it is false and the speaker meant that his dog is white.
It is not in physics, not in javascript, neither is it in morality, it is in grammar, therefore it is grammatically incorrect.
Ah, how conveniently you left out 'semantically'.
It is wrong semantically, as it uses the wrong meanings of the words. It is semantically wrong, but not grammatically wrong.
You keep evading:
"Bob is a splenetic guy" is grammatical even though the speaker misused the word 'splenetic' thinking it means what we mean by 'splendiferous'.
As to teaching English, of course it is needed not only to say that the sentence is false but that it is false because the words don't mean what the speaker thinks they mean. But that still doesn't make "My cat is black" ungrammatical. It is both (1) False and (2) False on account of the wrong words being used. But it is still grammatical.
When we consider whether an utterance is grammatical, we don't first check what the speaker meant by the utterance. We merely look at the words themselves. If I give you this:
"The cat is black" and ask, "is that grammatical?" You don't track down the speaker and find out whether he knows the definitions of 'cat' and 'black'.
My mistake about 'adjective'; I do know that it is an adverb.
But there's another example:
"'literally' is an adjective" is grammatical, even though false, and even though it is false by dint of the speaker using a word incorrectly.
And I'm merely talking about the fact certain parts of speech are required to be certain positions and in relation with other parts of speech. "Black the is cat beautiful' is not grammatical as the part of speech are not in correct order, but "The black cat is beautiful" is grammatical.
What is regarded as rational may be different for different people.
— TonesInDeepFreeze
I addressed that before, it is tangential:
If people have different concepts of rationality, then they may differ as to what laws of thought they adhere to, thus there are laws of thought that may be broken.
Doesn't matter what the definition is. People may break all kinds of norms of rationality in their thinking.
— TonesInDeepFreeze
Ok, clearly you are operating under a thin definition of rationality, where one even can think irrationality.
That is exactly what I am not saying. I am not at all saying that rationality permits irrationality. Rather, I am saying that people may break rationality, thus they may break a given law of thought. I even said that, of course, tautologically, adhering to rationality requires adhering to rationality.
Let's understand instead 'laws of thought' as the necessary conditions/operations for my/human/any rationality. Since they are necessary, they cannot be broken. If a mind does not obey them, that mind is no longer a (my/human) rationality.
That's okay. But it is different from saying that the laws of thought cannot be broken. If we consider those laws of thought to be necessary for rationality, then they cannot be broken without incurring irrationality. But they still can be broken.
The way it read was that there are laws of logic that may be broken but not laws of thought.
— TonesInDeepFreeze
Correct.
But if any law of logic may be also a law of thought, then there are laws of thought that may be broken too.
— TonesInDeepFreeze
Instead, if a law of logic can somehow holistically and correctly express a law of thought, that law of logic cannot be broken. If it can, it is not longer a law of thought, as by the definition I gave above.
But, if I recall correctly, you said that in general laws of logic can be broken, as you even gave an example of breaking the law of noncontradiction. Moreover, if there is a single law of logic that can be broken, and that law of logic corresponds with a law of thought, then there is a law of thought that can be broken. Moreover, even that point is not required, since we know that people do break laws of thought. Though, of course, if a certain law of thought is required for rationality then it can't be broken without incurring irrationality.
?Leontiskos I don't think there are laws of logic that cannot be broken, but that there are laws of thought that can't be broken (for obvious reasons). Some laws of logic may express those laws of thought. But that is just a semantic contention.
Everytime you say those well-formed phrases are syntactically correct, I agree. But they are not grammatically correct if the speaker thought/meant something other than what those words actually mean. So I cannot say they are grammatically correct.
Now, you're arguing by reiteration of your claim. When it comes full circle like that more than once, rational discussion is diminished.
When you quote people here, the original italics or bold are lost, so it is of common understanding that, when a quote features those, it is the quoter who has added them for a purpose.
What? You don't know how "[emphasis added]" works?
My original did not have bold. You added bold to my quote. When you do that, you should include a note that you added the emphasis. It is not up to the reader nor me to recall the peculiarities of the formatting processes of this site to then reason, "So the bold would have been lost if it were quoted, so if it appears, then it must have been added."
Whether it was three decades ago or three seconds ago, it is not proper to display someone's quotes with emphases they didn't use unless you indicate that the emphases were added.
: the study of the classes of words, their inflections (see inflection sense 2), and their functions and relations in the sentence
: a study of what is to be preferred and what avoided in inflection (see inflection sense 2) and syntax (see syntax sense 1)
: the characteristic system of inflections (see inflection sense 2) and syntax of a language
: a system of rules that defines the grammatical structure of a language
syntax
: the way in which linguistic elements (such as words) are put together to form constituents (such as phrases or clauses)
: the part of grammar dealing with this
/
Nothing there about semantics or meanings. Rather, the structural aspects.
Especially in logic and philosophy of language, usually 'syntax' and 'grammar' are understood together. And semantics is different. Syntax concerns whether an expression is well formed. Semantics concerns the meaning of the expression.
the whole system and structure of a language or of languages in general, usually taken as consisting of syntax and morphology (including inflections) and sometimes also phonology and semantics.
Not that I trust "Google dictionary", but you proffered it. So:
The usual sense of 'grammar' is 'syntax'. But sometimes it includes semantics. So I will award myself the point that usually 'grammar' and 'syntax' are used the same. I will award you the point that sometimes semantics is included. But consider that in logic, usually a sharp distinction is made between syntax and semantics and use of 'grammar' would align with 'syntax' not 'semantics'. The quotes you give do indicate a more extended sense of 'grammar'. I haven't seen that sense in logic or philosophy of language, but if you insist. Meanwhile, you could have easily ascertained that 'syntax' and 'grammar' are commonly used interchangeably but that your context is different.
Oh wait, the Google entry is just the Oxford entry, so as you posted it redundantly, I will too:
The whole system and structure of a language or of languages in general, usually taken as consisting of syntax and morphology (including inflections) and sometimes also phonology and semantics; grammar was one of the seven liberal arts.
— Oxford Reference
You can just click on the arrow to see what post the person is referring to instead of guessing.
I'm not talking about guessing what post was quoted. I'm talking about the fact that it is ridiculous to expect a reader to factor in the peculiarities of the formatting of quotes to know whether the emphasis was original or added.
A reader shouldn't have to click back to find out where the bolding came from. It is the responsibility of the quoter - not the reader and not the quotee - to indicate that the emphases were added.
My post there is from 3 hours ago. I was not reiterating anything.
You're reiterating your claim as you made it at the start of this round.
So, I'll reiterate:
"Jack is happy" is grammatical even when the speaker misused the word 'happy' while thinking it meant 'doleful'.
We don't have to ask the speaker what he meant to check whether he knows the correct meanings of the words. We just have to look at the sentence to see that it obeys the formation rules for the language.
And, in logic, which I hope was the original context, the usual distinction is between syntax and semantics, and with 'grammar' sometimes mentioned rather than 'syntax'.
Now we're full circle more than three times at least.
"The cat is black" and ask, "is that grammatical?" You don't track down the speaker and find out whether he knows the definitions of 'cat' and 'black'.
— TonesInDeepFreeze
Of course. It doesn't mean however that it was grammatically correct. We assume it is because we assume the speakers know how to use words.
No, we don't need to make any such assumption. You're just stipulating that out of thin air.
We might know nothing about who or what wrote an expression.
If I display a sentence on a piece of paper and leave it on the sidewalk, and you pick it up and read it, "The cat is black", then you recognize that as grammatical, no matter whether written by Shakespeare or a random word generating machine or an insane person who thinks 'cat' means 'screwdriver' and 'black' means 'wet'.
Suppose you have a job correcting school assignments, and you are never in the classroom, never met the kids, you just correct the papers. Then you don't know the vocabularies of the kids. You don't know which ones know the correct definitions of the words used. But you can still correct and grade the grammar. If you see, "The car engine is noisome", then you mark the sentence as grammatical, even though you don't know whether the kid knows that 'noisome' means 'noxious' and not 'noisy'.
Really, that is so plain that if you still refuse to understand it, then indeed you defy rationality.
Curiously, the BE article also has to take refuge in modern French words to express itself:
It's a nod to Saussure (and his demonic, hyper-nominalist post-modern semiotics of destruction—as opposed to our Augustine and Co.'s virtuous and sure triadic semiotics of life).
English-speaking philosophers who embrace contemporary French philosophy have to be very careful to use the French, lest they commit a massive faux pas like pronouncing Derrida's "différance" with a French accent. This is embarrassing indeed. The whole point is that différance and différence are pronounced exactly the same, and so one can only tell the difference when looking at them on paper (the victory of the logocentric over the phonocentrism of Saussure's Grammar, which has now been deconstructed).
However, the idea that signs might have something to do with a res or referent is terribly naive, if not downright provincial. "Intent of the speaker?" In grammar?
Friend, surely you know that both the author and the utterer have been dead for decades now? It would be totalitarian, not to mention brutish to suppose that intent should be allowed to tell people how they should construct their meaning, or that the properties of res/signified should play any determinant role in shaping the meaning/dicible in some sign relation. After all, people are themselves just signifiers, nexuses of sign-based discourses.
"Rob have a piink horn on his forhead", syntax is fine
The syntax is not fine. (1) 'have' should be 'has' (2) 'piink' is not a word (3) 'forhead' is not a word'.
Syntax checks. Are these words? Are the words in an allowed order based on their kind? Are the words in correct case, inflection, etc.
Semantics checks: What are the meanings of the words? What are meanings of the clauses? What is the meaning of the sentence?
Same with formal logic. Syntax cheks: Are these symbols of the language? Do the sequences of symgols form formulas? Are the sequences of formulas allowed as proofs according to the rules?
Semantics with formal logic: To what do the symbols refer? What are the truth values of the sentences based on the meanings of the symbols and subformulas?
You would say the first one is grammatically wrong, because 'criteria' is plural. Here is the problem: there are actually some people in the world whose first name is Criteria.
I can't believe you stooped to such a sophomoric argument. Obviously, we consider a context in which we at least agree as to the kind of word. Your argument is horrible desperation.
And still it doesn't answer that the "The cat is black" is seen to be grammatical even if the author of the sentence is anonymous, and even if we don't know whether the author is a human being or understands anything about language, as the expression could have been randomly generated and only by luck came out grammatical.
For about the dozenth time:
If you show me "The cat is black" then I will mark it as grammatical, not matter where you got the sentence.
But, if I recall correctly, you said that in general laws of logic can be broken, as you even gave an example of breaking the law of noncontradiction.
— TonesInDeepFreeze
Yes
Which brings me back to my point that is sustained:
But if any law of logic may be also a law of thought, then there are laws of thought that may be broken too.
— TonesInDeepFreeze
Instead, if a law of logic can somehow holistically and correctly express a law of thought, that law of logic cannot be broken. If it can, it is not longer a law of thought, as by the definition I gave above.
— Lionino
But, if I recall correctly, you said that in general laws of logic can be broken, as you even gave an example of breaking the law of noncontradiction. Moreover, if there is a single law of logic that can be broken, and that law of logic corresponds with a law of thought, then there is a law of thought that can be broken. Moreover, even that point is not required, since we know that people do break laws of thought. Though, of course, if a certain law of thought is required for rationality then it can't be broken without incurring irrationality.
I have edited the post you are quoting. So now it reads "as the necessary conditions/operations for my/human/any mind". In this sense, I don't think it can be broken, as the mind, definitionally, cannot operate outside of these conditions.
People operate mentally in all kinds of ways: Fictionally, absurdly, poetically, ironically, day dreaming, dreaming, mystically and insanely. But your point reduces to the tautological: the mind can't operate rationally without operating rationally. No one disagrees with that.
I still think the LNC overall articulates a law of thought
That is not at issue. What is at issue is whether that law of thought can be broken. Yes it can. Of course, if we hold that it is required for rationality, then we may say it can't be broken rationally. But that doesn't refute that people break laws of thought often. Full circle again.
Merriam Webster is not reliable neither is it competent.
I have found Merriam to be good, especially unabridged, but some deterioration over the years. I read all of yours. I mentioned the others for emphasis.
The usual sense of 'grammar' is 'syntax'
— TonesInDeepFreeze
It is not.
From definitions you posted yourself.
/
If I were to nitpick, it would be a whole other thing. Being careful to state things about logic accurately so that false conclusions about are not drawn is not nitpicking.Reply to Deleted user
It is a misspelt word. It has nothing to do with syntax.
In logic, terms are formed by rules. If symbols are not in correct order or incorrectly omitted, then they are not syntactical. In that way too, if a string of letters doesn't even form a word, then the expression in which the string occurs cannot be syntactical.
every sentence is grammatical. Not every sentence is grammatically correct.
Oh, please! Talk about inane nitpicking that isn't even correct! Obviously I'm using 'grammatical' in the sense of 'conforming to the rules of grammar'.
One instance that I can see might be regarded as nitpicking was when I said saying "B is true" was extraneous. But I mentioned it in a stylistic sense that it's better not to include extraneous items so that the arguments can be seen more clearly, without the distraction of those items.
Right, I was just pointing out that this is almost certainly what the reference to "in modern theory," was referring to. This is how the French makes it into English sources.
Saussure is relevant to the conversation at hand in that his later post-structuralist disciples eventually worked themselves towards totally divorcing meaning from authorial intent and context. And this move was given an almost political connotation, a "freeing of the sign." Although one might question if some of the further evolutions of this way of thinking might not just succeed in freeing language from coherence and content.
I think there is actually a connection here to how formal grammar is conceptualized. In either case, the focus becomes signs' relations to other signs, pretty much to the exclusion of context or content.
But your point reduces to the tautological: the mind can't operate rationally without operating rationally. No one disagrees with that.
— TonesInDeepFreeze
I am aware of that. The tautology therefore is about law of thought, not about laws of logic, a different concept, thus it does not follow that laws of logic are unbreakable.
Yes, it doesn't follow. No one said otherwise. And yes, I was referring to your notion of the laws of thought. I'll say it again:
One can break the laws of thought on pain of being irrational. But you say that the laws of thought are unbreakable. But one can break the laws of thought. So you regroup by saying that one can't break them and be rational. But that is not at issue. My point is that one can break the laws of thought, contrary to your earlier claim.
if there is a single law of logic that can be broken, and that law of logic corresponds with a law of thought, then there is a law of thought that can be broken
— TonesInDeepFreeze
If the law of logic is understood as expressing a law of thought — which in modern days that is not how it is understood
Where is there a report that modern writers in general believe that laws of logic may not be understood as expressing laws of thought? And what period do you regard as modern?
hence my original comment to Leontiskos —, by definition it can't. If law of logic is understood as how we understand it today, laws of thought do not correspond to laws of logic because, as we have agreed, the latter may not be respected by some system, they may only allude to or be based on laws of thought.
I'm uncertain whether I understand you. Certain systems don't respect certain laws of thought. That doesn't entail that laws of thought cannot be broken. Indeed, it evidences that they can.
Also, you say "the latter", which is 'laws of logic'. So 'they' also refers to 'laws of logic'. And you say 'they may only allude to or be based on laws of thought'. So that is saying that laws of logic may only allude to or be based on laws of thought. But that seems the opposite of anything we've agreed on. If the laws of thought require rejecting contradiction, then systems that allow contradiction do not adhere to that law of thought.
I'm not talking about guessing what post was quoted.
— TonesInDeepFreeze
I am. You constantly [emphasis added] mistake what post is being quoted.
(1) In one case, I was unclear as to whether you were quoting in agreement with the quoted poster. And I overlooked that your recent lashing out was not directed at me. That is not even remotely constant (2) In this instance, I've been in exactly the right place about what was posts was referenced.
"Jack is happy" is grammatical even when the speaker misused the word 'happy' while thinking it meant 'doleful'.
— TonesInDeepFreeze
I have refuted that already. Talking of circles.
Your replies don't even come close to a refutation.
It's plain as day: One can easily see that "The cat is black" is grammatical, without having to know anything about the person who said it, or even if it was not said by a person but formed randomly by a machine. You've not refuted that. One of your replies is that we assume the speaker knows the meanings of the words. But that is not necessary to see that the sentence is grammatical. We could say, "I have no idea whether the person who wrote "The car engine is noisome" knows that 'noisome' means 'offensive' not 'noisy' but that doesn't matter if all you want to know is whether the sentence is grammatical. I'll happily and without any reservation tell you that is."
/
Oh, and about nitpicking: Your objection to "If ___, then ___" is a doozy!
People operate mentally in all kinds of ways: Fictionally, absurdly, poetically, ironically, day dreaming, dreaming, mystically and insanely.
— TonesInDeepFreeze
And all of those operations are operations of the mind, therefore bounded by the rules of the mind, which we may call laws of thought.
You just completely ignore the point, that I've made twice, now a third time:
In such mental states, people often break the laws of thought.
The refuted person may not be disposed to accept that he's been refuted. But it doesn't follow that if a person points out that he's not been refuted (and gives clear argument about that), then that person is doing that because he doesn't want to admit to having been refuted.
I thought they were two different definitions. But the second includes additional assertions beyond what I would have thought is a definition. Also, I don't know what 'instead' refers to.
A law of thought is necessary for the mind no matter what it is doing, ironising, dreaming, thinking, or whatever. All of these have subjacent operations that are necessary to them.
Whatever is "subjacent", in those mentioned mental states, the laws of thought are broken in the sense of irrational thinking, believing or imagining. If a mystic experiences contradictions as being true, then he's not breaking the laws of thought? If one dreams that one's great-grandfather is both alive and dead at the same time, one is not breaking the laws of thought?
I'll try to combine your clauses into a defintion:
Laws of thought are facts about your mind such that those facts are necessary for the operation of the mind.
I don't know if that's what you mean, but it's my best guess.
Or maybe just say:
Laws of thought are the necessary mental conditions for the operation of the mind.
From that definition, it follows that they can't be broken.
/
So, when a person is utterly irrational, they are still obeying the laws of thought on account of the fact that there are mental conditions necessary for the operation of their mind?
I didn't mention you skills. I mentioned your knowledge.
And you don't have to feel they that my view is needed nor do you have to request it for me to state it.
Meanwhile, you lashed out at another with your characterization of his knowledge of language. Same applies to you in your knowledge of logic. You've made hundreds and hundreds of posts about logic that are a dead end as your gravamen can be neatly summarized in a couple of sentences (as I did for you) without the pointless variations all on the same pointless theme.
I don't require your courtesy. And I don't require you not to post so that you don't wear out my patience as you do. Anyway, in general, many people in this forum will be discourteous quite soon after they are disagreed with.
[EDIT: "courtesy" from a guy who makes a ridiculous argument against the common courtesy of noting that emphases were added to a quote.]
You can post or not post as you please. And I'll do the same.
I don't pretend to be a bully and I'm not one. And "senile" is to guffaw.
Meanwhile, no matter how you regard me as "coming off", I don't manufacture perceptions about you in that way. No matter how you "come off" to me, I regard the substance of your posts, good or bad, on their own terms, not personally.
If a mystic experiences contradictions as being true, then he's not breaking the laws of thought?
— TonesInDeepFreeze
I don't think any such experiences are possible.
Of course they're possible. Whether in absurdist day dreaming, insanity, dreaming or in mystic state, one can have all kinds of irrational thoughts and dispositions. Even in everyday experience, people often drift to sleep with disconnected nonsensical ideas and irrationality.
But, if it is the case that it is possible, definitionally there are no laws of thought that preclude from that happening, because it happened, therefore oen is not breaking laws of thought.
Yes, and therefore "laws of thought" pretty much reduces to simply "conditions necessary for mentation". If whatever one thinks, no matter how irrational, is not breaking the laws of thought, then the notion of 'laws of thought' is so general that it is hardly worth mentioning. That suggests putting some more meat on the bones of your definition.
if by irrational you mean things of the sort of believing the colour green is sweet and that the moon is made of cheese.
Synesthesia does occur. And people have all kinds of false beliefs not derived by good inferences. But beyond those, people also have even more profoundly alternative states.
I should not have honored that garbage even by laughing at it.
"senile" is juvenile. Worse, it's pernicious. One would think that such crude ageism wouldn't get into public past the lips of a putatively aware poster. People have mental difficulties for many different reasons. It's not a matter of age, but of the difficulties no matter their cause. Meanwhile, bigoted ridicule of people for their age is obnoxious and disgusting. Also pretty bad is to compound that bigotry by making it a term of general insult against targets whose age is not even known and not relevant no matter what it is.
Although one might question if some of the further evolutions of this way of thinking might not just succeed in freeing language from coherence and content.
Ok, so your "A without B" is not that "it is possible to have A without B", but that "there is A without B". I guess that can make sense as ¬(A?B) ? (A?¬B).
The oddity is that there is not parity between a conditional and its negation:
A conditional, by its very name, signifies that which is not necessary. (1) is therefore conditional in that it neither commits us to A, ¬A, B, or ¬B. It retains something of the hypothetical nature of natural-language conditionals.
(2) is not a conditional in this sense, for it commits us to both A and ¬B.
In natural language when we deny a conditional we at the same time assert an opposed conditional; we do not make non-conditional assertions. In natural language the denial of a conditional is itself a conditional. But in propositional logic the denial of a conditional is a non-conditional.
As noted in my original post, your interpretation will involve Sue in the implausible claims that attend the material logic of ~(A ? B), such as the claim that A is true and B is false. Sue is obviously not claiming that (e.g. that lizards are purple). The negation (and contradictory) of Bob's assertion is not ~(A ? B), it is, "Supposing A, B would not follow."
Given material implication there is no way to deny a conditional without affirming the antecedent, just as there is no way to deny the antecedent without affirming the conditional.
You are thinking of negation in terms of symbolic logic, in which case the contradictory proposition equates to, "Lizards are purple and they are not smarter." Yet in natural language when we contradict or negate such a claim, we are in fact saying, "If lizards were purple, they would not be smarter." We say, "No, they would not (be smarter in that case)." The negation must depend on the sense of the proposition, and in actuality the sense of real life propositions is never the sense given by material implication.
"If lizards were purple then they would be smarter."
The denial is, "Even if lizards were purple, they would not be smarter." It is not, "Lizards are purple and they are not smarter." The logical negation is the English counterexample.
---
The deeper issue here is that there is no uncontroversial way to translate between English and formal logic, because English has inherent meaning where logic has none. Logical meaning is derived from English meaning, and not vice versa. Because of this the "meaning" of a logical sentence is merely what can be done with it, or what it can be transformed into, and no one transformation is more central to its "meaning" than any other. This is what I was trying to get at on the first page.
Bongo did a good job of using English to capture the range of the logical possibilities, but at least one problem arises in that the English negation and the logical negation are substantially different. As you pointed out in the other thread, a central aspect of an English negation of a conditional is that the consequent and only the consequent is negated (e.g. If <lizards were purple> then NOT<lizards would be smarter>).
(What this then means is that to unequivocally claim that Reply to flannel jesus' scenario does not represent a contradiction is to rely exclusively on a "bug" of material implication, and only those who are able to contextualize material implication within a larger whole will be able to consider the question more fully.)
Saying «A implies B» is A?B, but «A does not imply B» doesn't take the ¬ operator anywhere.
Yes, for it is not possible to capture the negation of the idiosyncrasies of material implication while simultaneously capturing the negation of the notion of implication or conditionality. One or the other must be lost. English abandons the first and propositional logic the second.
Reply to AmadeusD - Perhaps. I am thinking of the example that Janus gave elsewhere.
P: Lizards are purple
S: Lizards are smarter
(P?¬S)
"P does not imply S"
I think the English sense is never falsified by the logical sense, and in that way it would seem to hold. The problem is that the logical sense can be falsified by alternative English senses, given that English has a more robust notion of implication than material implication. So you can't go in the other direction. Ergo, you cannot translate (P?¬S) as, "If lizards are purple, then they are (necessarily) not smarter," even though you can draw the conclusion, "P does not imply S" ("Lizards' purpleness does not imply lizards' smartness").
Edit: So we might say that (1) guarantees (2) but (2) does not guarantee (1). Thus I admit that it doesn't count as a real translation.
Edit2: I think Janus' argument is special insofar as it makes use of a Cambridge property, and in that case (1) and (2) seem to be the same.
So then why is it that the logic cannot capture the English, "A does not imply B"? Is it because the English represents a denial without any corresponding affirmation?
If so, it seems that I was wrong when I said that to deny a conditional in English is necessarily also to affirm an opposed conditional:
In natural language when we deny a conditional we at the same time assert an opposed conditional
In English we can deny in a manner that does not affirm the negation of any proposition, and this violates the way that propositional logic conceives of the LEM. In fact, going back to flannel’s thread, this shows that a contradiction in English need not take the form (A ^ ~A). In English one can contradict or deny A without affirming ~A.
...but then again maybe to say “Not A and not ~A” is only open to Buddhist-type logic or Buddhist-type English. Even if that is so, perhaps what is available more broadly is the denial of a consequence without any attendant affirmation, such as, “That does not follow from this, and I make no claim about what does follow,” as I claimed <here>. In this way one undercuts an inference and deprives the conclusion (or consequence) of its validity without falsifying the conclusion. Thus one can say, “A does not imply B,” without making any positive assertion, conditional or otherwise. Apparently the relation between a negation and an affirmation differs in English and in logic.
Edit: This may actually be key to understanding A?(B?¬B), for the contradiction is nonsensical or unstable when taken in a particular sense, and this is why the standard logical operations cannot be applied to it in the same way. A reductio ad absurdum may be parallel to the English move of denying a conditional without affirming anything in the same move. If a reductio affirmed something in the same move then there would be no and-elimination step, and if that were so then a reductio would be identical to a modus tollens, which it is not. The affirmation involved as the final step of a reductio only takes place "after" the and-elimination step. The contradiction is repugnant regardless of which conjunct is preferred (or of which supposition was originally made), and this makes sense because what is proximately aimed at in a reductio is contradiction per se—a universal concept—rather than the application of any truth value to a variable. The application of the truth value to the variable is what is remotely aimed at, and will only take place after the contradiction and the and-elimination have already occurred.
My conclusion thus far is that «A does not imply B» can't be translated to logical language. I attempted several different ways in flannel jesus' thread but none worked.
Couldn't it be said that logical language establishes a number of precise connections between states, but the absence of a connection is not defined. It's the negative space that remains outside the ruleset.
then it would seem that we don't intuit negation in this case as a photographic negative of the Venn diagram, which is what logic would deliver. In which, i.e., all previous no-go (shaded) areas are declared open for business, and all previous open regions are shaded out. Rather, the intuition is that a (in this case the) previous no-go area is opened up. But nothing closed off. We wish to withdraw or deny an assertion without thereby committing to its negative. Deny it is the case there won't be a sea battle, without claiming there will.
So, not really negation. Not cancelling out the first. Not restoring not(A without B) to A without B.
¬(A?B) appears to suggest, intuitively: maybe A without B, maybe not. No commitment. No information. Tautology. No shading in the Venn diagram. (Whose 4 non-overlapping areas correspond to A & B, A & ¬B, ¬A & B, ¬A & ¬B.)
I am starting to think that it is because the word "implies" has the idea of causality in it, while logic says nothing about causality. I reckon that it is better to think of a truth table as coexistence rather than causation.
Yep, I think this is right, and it's what I was trying to get at on the first page. I think my point about "denying without affirming a propositional negation" is also right, and Bongo developed that point. I wonder if the two can be brought together.
and use instead "not A without B", which is exactly understood in English as coexistence.
In general I want to avoid thinking any English represents the logic, but I also I think this is a good point. But to give something of a counterexample, if A is false then we can say A?B, and yet your English does not capture this move. Thus:
Because of this the "meaning" of a logical sentence is merely what can be done with it, or what it can be transformed into, and no one transformation is more central to its "meaning" than any other. This is what I was trying to get at on the first page.
Keeping to this counterexample, "not A without B" captures a truth-functional conditional, but it does not fully capture a material conditional. English involves "causation," but it additionally prescinds from the idiosyncrasies of the material conditional. I think Bongo's negation may have more to do with the materiality of the conditional than its lack of causation, although the two may well be related.
You mean that saying "He is not beautiful" is not necessarily the same as saying "He is ¬beautiful"?
Ha - that's an additional consideration that I was not thinking of (Diotima's point in the Symposium). Prescinding from this question and from the question of Buddhist logic, my point is primarily about conditionals or consequences, and can be set out in response to Bongo:
We wish to withdraw or deny an assertion without thereby committing to its negative. Deny it is the case there won't be a sea battle, without claiming there will.
Basically, but more precisely, I would say that we are denying an inference. In English we don't usually say, "You are wrong that there will be a sea battle tomorrow, and yet there may be a sea battle tomorrow."* Instead we say, "Your reasoning for why there will be a sea battle tomorrow is not sound, and yet there may be a sea battle tomorrow."
N: There is a wind from the north tonight
S: There will be a sea battle tomorrow
N?S
The denial in English would seem to be, "S does not follow from N." This doesn't mean that S will always be false whenever N is true. It only means that S need not be true when N is true. This seems to be evidence for Lionino's view that a causal connection is at play. Or as I said on the first page, "The English has to do with a relation between P and Q that transcends their discrete truth values."
then it would seem that we don't intuit negation in this case as a photographic negative of the Venn diagram, which is what logic would deliver.
...
So, not really negation. Not cancelling out the first.
These are good thoughts, but I think a kind of cancelling-out is taking place. It's just that the denial transcends the limitations of truth-functional logic.
To deny something requires understanding what is first being asserted, that it might be denied. If someone says, "Wet grass follows from rain," they are not asserting everything that is involved in the logical claim <[rain]?[wet grass]>, for they are not asserting the idiosyncrasies of the material conditional, such as the idea that ~[rain] justifies their claim. At the same time, they are asserting something more than the logical claim insofar as they do not believe that the falsity of their claim would mean that rain always produces dry grass. Something more subtle is being said and something more subtle is then in turn being denied, and these subtle affirmations and denials don't straightforwardly translate into the affirmations and denials of classical propositional logic.
Or going back to my earlier post and putting it in simpler terms, we can deny a conditional with a simple denial of the metaphysical relation, or else with a counter-conditional, or else with a counterexample. When classical propositional logic denies a conditional it is limited to doing so with a counterexample (e.g. N ^ ~S). This is something of a bug, for to deny the essence of a conditional is to deny its conditionality (e.g. "N does not ensure S"). English is capable of all three responses; propositional logic is only capable of one.
*I am changing the proposition to avoid confusing double-negatives.
Yes, the red and white system at least. Unfortunate that it shades in where I was shading out. But it shows how logic uses "not" as a reversal of shading, sending anything in row 2 to row 4 (and vice versa, and also reversing shading within row 3). Whereas ordinary language, while it might do that, might equally well signal a retreat to the very top, leaving all options on the table.
Or (@Leontiskos) it might do something else more elaborate which deserves analysis. Rabbit holes galore, of course. :grin:
I am starting to think that it is because the word "implies" has the idea of causality in it, while logic says nothing about causality. I reckon that it is better to think of a truth table as coexistence rather than causation.
So I think you are overstating this idea. Conditionals have a directionality that partially mimics causality. Meta-logically, they are intended to support the inferences of modus ponens and modus tollens. These are directional, asymmetric inferences. When you think in terms of coexistence or when @bongo fury speaks about "Not A without B" or Venn diagrams, you both seem to be thinking primarily in symmetric, non-directional terms. For example, "coexistence" is not asymmetrical or directional like (A?B). It is symmetrical like (A^B).
Further, "A implies B" does not necessarily mean that A causes B. As I said on the first page, the relation can be indicative. For example, the antecedent can be a sign of the cause that is the consequent. "Warmth implies fire," does not mean that warmth causes fire. "Implies" can also be correlative, where two correlates are caused by a third thing, but this is a true case of coexistence, in which the relation is biconditional (and therefore symmetric) rather than merely conditional (and therefore asymmetric).
There are lots of legitimate ways to speak about (A?B) in English, and each is incomplete:
Forms relating to ¬¬(A?B):[list]
"Not(A without B)"
"Not A without B"
"No A without B"
"¬A and/or B"
[*]Forms relating to modus ponens:
"If A then B"
"A implies B"
"B follows from A"
"B from A"
[*]Forms relating to modus tollens:
"If ¬B then ¬A"
"¬B implies ¬A"
"¬A follows from ¬B"
"No ¬B without ¬A"
"Without B, no A"
[*](I omit the forms relating to the idiosyncrasies of material implication)
[/list]
There are also lots of legitimate ways to speak about ¬(A?B) in English, and each is incomplete:
Forms relating to (A^¬B):[list]
"A without B"
"A but not B"
"A and not B"
[*]Forms relating to the denial of modus ponens:
"Not(If A then B)"
"A does not imply B"
"B does not follow from A"
"No B from A"
[*]Forms relating to the denial of modus tollens:
"Not(If ¬B then ¬A)"
"¬B does not imply ¬A"
"¬A does not follow from ¬B"
"¬B without ¬A"
"B requires no A"
[*](I omit the forms relating to the idiosyncrasies of material implication)
[/list]
Of course some of these overlap. For example, the multiple meanings of "without" make "Not A without B" ambiguous between a directional modus ponens and a non-directional ¬A?B.
Again, "No one transformation is more central to [the logical] 'meaning' than any other" (Reply to Leontiskos). Privileged meanings only emerge at the meta-logical level:
On the other hand, in English, or most European languages, nobody ever says "X implies false/true", that comes off as gibberish. The reason must be because the word 'implies' has the sense of (meta)physical causation, while logical implication is not (meta)physical causation; the latter starts with the antecedent being true, the former may have a false antecedent.
If it rains, then the grass will be wet.
If Hitler was a military genius, then I'm a monkey's uncle.
These are equivalent at the first-order level, but not at the meta-logical level. At the first-order level they are both true and there is no difference between the truth of (1) and the truth of (2). At the meta-logical level, (1) partakes in the true purpose of a conditional whereas (2) does not (link). (2) is a consequence of the idiosyncrasies of the material conditional. This relates to my earlier point that the logical negation of a conditional is no longer a conditional, and in that case the modus ponens and modus tollens are no longer accessible, and because of this the directionality of the conditional dissipates.
If one does not make the meta-logical distinction between (1) and (2) then they will be tempted to claim that conditional logic cannot map asymmetrical or directional relations (including causation). This isn't right. A conditional can map an asymmetrical relation. Can it map something like causality? Yes and no: partially but not fully, because causation is not entirely truth-functional.
The key here is that propositional logic distinguishes (1) from (2) not in themselves, but extrinsically through modus ponens and modus tollens. Even though (2) is 'true', nevertheless it cannot be used to draw any substantial conclusion. Calling the conditional "true" is just a useful fiction which has no practical impact on the system. Or rather, it shouldn't. In the other thread we are seeing the havoc that meta-logical ignorance can wreak, for to permit standing contradictions gives the "dross" of the material conditional a potency it was never intended to have. It turns the useful fiction into non-fiction.
Of course some of these overlap. For example, the multiple meanings of "without" make "Not A without B" ambiguous between a directional modus ponens and a non-directional ¬A?B.
I suppose it is worth asking whether these are the same two inferences, and whether the first is any more "directional" than the second:
(A?B)
A
? B
¬A?B
A
? B
I want to say that they are different mental inferences, which is why we think of them differently (in English). But this is part of the difficulty of the thread. "Rabbit holes."
But it does. If we understand A?B as «not A without B», and we have ¬A, it is within the scenarios that «not A without B» precludes, because it only precludes A, ¬B, it doesn't preclude ¬A ever.
I think you may have mixed up a bit of the verbiage there, but I think you are saying that «not A without B» prescinds from whether or not ¬A justifies the conditional, and that is precisely my point. «not A without B» does not capture the fact that ¬A makes the conditional to be automatically true.
Or in other words, I can say, "¬A, therefore A?B," and clearly «not A without B» does not justify such a move. If all we knew about A?B was «not A without B», then we would not know that such a move is valid.
What I am saying is that knowledge of (2) does not give us knowledge of (1), and yet everyone who knows what A?B means has knowledge of (1). Therefore (2) does not give us complete knowledge of A?B. (2) does not fully represent A?B.
(Edit: I am pointing to a problem with your claim that we can translate A?B into English as "Not A without B.")
However, what about ¬(A?B)? What can we say about this in English? The first thought is "A does not imply B". But here is the trouble:
if ¬(A?B) is true
and B is false,
A is true.
No, your conclusion (A is true) is not valid. You seem to be interpreting “¬(A?B)” as: “¬A->¬B”, and that’s invalid. “¬(A?B)” just means that the truth value of A does not give us a clue as to the truth value of B. A better English translation of ¬(A?B) is : it is not the case that A implies B
Consider these substitutions:
A=All bluebirds fly
B=Fred is a duck
This is consistent with ¬(A?B) being true. If we discover Fred is a pigeon then B is false, but it tells us nothing about whether or not all bluebirds fly.
The question is not whether ¬A is allowed, but whether ¬A ? A?B.
<"Not A without B" does not preclude ¬A> is a different proposition than <If ¬A is true, "Not A without B" is true>.
¬A ? A?B
¬A ? "Not A without B"
(1) is true. (2) is false. It is false for you to claim that the consistency of ¬A and "Not A without B" justifies (2). (2) requires more than consistency. It requires more than that ¬A is allowed. "¬A is allowed, therefore (2) is true," is an invalid claim.
Put differently, we can know from «not A without B» that ¬A is not disallowed, but we cannot know that the statement is made true by ¬A.
You can infer A from ¬(A?B) by De Morgan.
¬(A?B)
¬(¬A?B) (definition of material implication)
¬¬A?¬B (de Morgan)
A?¬B (double negation)
I concede your point, but what you have proven is that:
¬(A?B)
Implies A
(Which I confess seems counterintuitive - see below*).
You had said: If A does not imply B, and B is false, A is true
That second premise(¬B) is superfluous to the conclusion (A).
--------------------------------------
*Now suppose we apply the logic to these statements:
A=All bluebirds fly
B=Fred is a duck
¬(A?B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")... which is certainly true because the antecedent has no bearing at all on the consequent
(¬A?B) = "not all bluebirds fly" or "Fred is a duck"
...
A?¬B: All bluebirds fly and Fred is not a duck
Problems:
Despite the fact that ¬(A?B) is a true statement...
1) it is NOT true that all birdirds fly (hatchlings don't fly),
2) My pet duck is actually named Fred.
But the logic conclusion says otherwise.
Something ain't right. I had to dig out my 1973 Logic textbook to understand the problem, but I'd like to see if anyone can identify the problem on their own.
I said 'allowed' there to simply mean true no matter the truth value of the other variable. If ¬A is not disallowed, it means it is true. ¬A is simply A is false or 0. Not A without B means that A=1,B=0 is false, therefore every other combination of the values of the variables gives us true. Since A=0 in the case that ¬A, not A without B is true, and so is A?B.
Ah, okay, I see where you are coming from now. It seems like a strange interpretation:
Aaron: "Not A without B"
Benjamin: "Not A"
Caleb: "B"
Daniel: "A"
Ephraim: "Not B"
Gregory: "C"
Frank: "It looks like everyone is in perfect agreement with Aaron, except for Daniel."
In English it is usually different to say, "Not A without B," and, "Anything which is not A without B is true."
Moreover, A?B does not follow from Ephraim or Gregory's answers in the way that «not A without B» does, and Daniel's answer seems to falsify «not A without B» without falsifying A?B.
I think they are there implicitly in "not A without B" as spoken.
This is why I would prefer "No A without B." The "parentheses" (however one wishes to depict them) become more important when you want to transform the proposition logically, or draw a modus tollens, etc.
I didn't really understand the Taleb-Nephlim dialogue but Daniel is just saying A but without saying anything about the value of B.
Sure, and in English is to say A without saying anything about the value of B to say A without B? It would seem so.
Would anyone interpret "Not A without B" as A?B unless they knew ahead of time that they were supposed to interpret it that way? It seems highly doubtful.
¬(A?¬B) is also no A without B. It says that A=1, B=0 is false.
The technical problem here is that the English "Not A without B" in no way circumscribes the domain as ((¬)A, (¬)B) pairs. Neither Benjamin, Caleb, Ephraim, or Gregory are saying A without B, and yet only Benjamin and Caleb's answers entail A?B.
For example, the only way to claim that Gregory's answer does not entail "Not A without B" while Benjamin's does, is to beg the question and assume that "Not A without B" is equivalent to A?B. Without that assumption there is no reason to think it is correct that ¬A ? «Not A without B» and incorrect that C ? «Not A without B».
¬(A?B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")
— Relativist
is not true.
The statement "It is not the case that ("all bluebirds fly" implies "Fred is a duck") IS true. But you're right that it's not equivalent to :
¬(A?B)
But why isn't it? It's because there is no material implication. The formula (A?B) cannot be used in all semantic instances of "if A then B".
I don't think I ever realized this before. When I took sophomore logic (50 years ago!), we concentrated on formulaic proofs. But the mapping to semantics is critical.
Sorry, this whole Benjamin thing is too confusing for me to keep up.
It is supposed to be simple: Has Benjamin agreed with Aaron? Has Caleb? Has Daniel?...
Or else: Aaron gives the condition, "Not A without B." Have the others fulfilled that condition or failed to fulfill it? The most obvious fulfillment would be, "A with B."
My point is that even the 2% who interpret it as, "No A without B," don't quite know what they mean by that. The real translation in those terms is something like, "No A without B in the domain of A-B pairs." Things like 'C' or '¬B' give no A without B, but they fail because they are not in the form of A-B pairs. Things like '¬A' or 'B' succeed when they are implicitly placed into the context of A-B pairs.
The point here is that if we sit down and think about what "No A without B" means in English, without assuming ahead of time that it means A?B, then we will recognize that it does not mean the same thing that A?B means. In some ways it does and in some ways it doesn't. My counterexample that began this whole tangent shows one of the ways that it doesn't.
at least some 1 in every 50 people would interpret it as no A without B
If the idea here is, "It's not necessarily a good translation, but it's the best we have," then I would ask why it is better than the standard, "If A then B"?
I think A?B is better translated as, "If A then B."
¬(A?B) is suitably translated as, "A and not B."
The logical negation of A?B is different than an English negation, for the logical negation is more intuitively a negation of ¬A v B, which goes back to <this post>.
If one wants to make the negation translatable into English then "No A without B" is perhaps the best candidate, but it is not the best candidate apart from that single motive. Again, in propositional logic the negation of a conditional is never anything more than a counterexample, and this is the bug we are dealing with.
However, what about ¬(A?B)? What can we say about this in English?
The solution you have arrived at is the idea that ¬(A?B) means, "A without B," and therefore (A?B) means "Not(A without B)." This misplaces the negations, acting as if the second negates the first when the opposite is true. What you are really saying is that ¬¬(A?B) means "Not(A without B)," and that (A?B) and ¬¬(A?B) are linguistically interchangeable.
What is really happening?
A?¬B means "A without B"
¬(A?¬B) means "Not A-without-B"
Then:
(A?¬B) ? ¬(A?B)
¬(A?¬B) ? (A?B)
And then you assume that the '?' is applicable not only for logic, but also for English, thus:
The problem was isolated in <this post>. A?B and ¬(A?¬B) (or ¬A?B) are not the same sentence. A?B directly supports relations like causality, whereas the other two do not. Further, the only way to prove A?B from ¬(A?¬B) is via an indirect proof such as RAA, which is an equivalence and not a derivation. "If P then Q," and, "Not A-and-not-B" are two different claims, both in logic and in English.
-
We can see this with an example.
A: I stop eating
B: I lose weight
The implication form is A?B ("If I stop eating, then I will lose weight"). This describes a relation between eating and weight. It means that to stop eating leads to losing weight, and that if one is not losing weight then they have not stopped eating (modus tollens).
The conjunction form is ¬(A?¬B) ("It is not the case that, it is true that I stop eating and it is false that I lose weight"). This says that A and ¬B cannot coexist. There is no relation posited between A and B.
The relation can be inferred from the conjunction, but it is not the same as the conjunction:
¬(A?¬B)
__Suppose A
__? B
? A?B
(4) follows from (1) and (2), but it is not equivalent to (1), despite the fact that the truth tables are the same. Put differently:
¬(A?¬B)
A
? B
[Meta-step: ¬(A?¬B), A ? B. Therefore, A?B given ¬(A?¬B)]
In your opinion arrows do not connote directionality? Do you think there is a reason logicians introduced the inference A?B over and above the conjunction ¬(A?¬B)?
"2% of the population might interpret ? as 'dragon', but that doesn't make for a very good translation". You see how that doesn't work?
Are you not equivocating between language speakers and non-language speakers? If only 2% of native speakers interpret ? as what we mean by 'dragon' in English then yes, it is a bad translation.
"No A without B in the domain of A-B pairs."
— Leontiskos
That is already implied by the phrase.
You think the English phrase, "No A without B," implies that we must be thinking about the entire domain of speech in terms of A-B pairs? This seems clearly incorrect. In English when we say, "No pizza without heartburn" we in fact order a salad ("C"), and this satisfies the condition just like Gregory's answer does.
It is saying there is no A, if there is no B. From A?B, ¬B, we infer ¬A — (A?B),¬B|=¬A. From A?B, C, we infer nothing about A because the value of B hasn't been declared. From A?B, C, ¬B, we infer ¬A, because C doesn't interfere — (A?B),¬B, C|=¬A.
You are again conflating the logic with the English. To think that the English entails whatever the logic entails is to beg the question and assume that the English perfectly maps the logic. That is what we are considering, not what we are assuming.
Regarding the modus tollens, the English does support it but, again, this is not the same as whether ¬A entails the truth of the conditional. These are not the same thing:
(A?B),¬B |= ¬A
¬A |= (A?B)
Specifically:
(A?B),¬B |= ¬A
¬A |= (A?B)
«No A without B»,¬B |= ¬A
¬A |= «No A without B»
Does (4) hold? It is questionable, but if it doesn't then the translation limps, and if it does then this also holds: < C |= «No A without B» >, in which case the translation also limps since C does not semantically entail (A?B). Either way the translation limps.
I think you took my "everything else is allowed" to mean literally everything else (C), but I meant "every other values of A and B".
But that's not what the English means. It is an arbitrary restriction of the English meaning. After all, if it's not being interpreted in favor of its literal meaning, then what is it being interpreted in favor of?
Part of the puzzle here is that in reality negations always obtain within a scope. For example, if C=salad, then C=¬A (not pizza). When we are within the same scope, C must always be either A or ¬A, and since C=¬A, C |= (A?B).
(Propositional logic seems to assume, prima facie, not only the commonsensical idea that C is neither A nor B, but also the deeply counterintuitive idea that C is neither ¬A nor ¬B. Usually if C is neither A nor B then it must be both ¬A and ¬B.)
One of them was the matter of putting logical formulas into natural language (English in our case) — that matter was essential for the purpose of correctly interpreting some statements.
A paper that shows how Medieval Aristotelian logic was in some ways more robust than current logic is Gyula Klima's, "Existence and Reference in Medieval Logic." He uses Russell's King of France example rather than conditionals.
Comments (262)
I have to agree that statements like A?B are universal, and I guess it helps people use logic quickly and easily. But, again, it is outstanding to see those logic formulas explained in language. It seems they are only allowed to use it with "A" and "B" in the premises.
Quoting Deleted user
I don't get it, but I'm confident I could get it using natural English. Is there a substantial difference?
Quoting Deleted user
To what extent should it be converted into logic formulas?
[i]All humans are mortal.
Socrates is human.
Therefore Socrates is mortal.[/i]
If you want to express in logic statements about logic itself -- which is a requirement for philosophical statements -- you even need to add support for arithmetic.
The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic.
I'd push back against this - this is one of the most egregious examples of logic disagreeing with our intuitive use of implication.
In classic symbolic logic, a -> b is true, according to its truth table, if, for example, a is true and b is true.
(2+2=4) implies (Kamala Harris is a presidential nominee). These is true in classical logic. But it doesn't really match our intuition at all.
"(2+2=5) implies (Kamala Harris is prime minister of China)" is also true in classical logic.
It actually does.
It just means that knowledge as a justified true belief is not only about truth but also about justification.
Maybe it matches your intuition, and I'm sorry for trying to speak for you. My mistake.
It is probably a mixup between the implication, which is just a truth table, and the entailment, a ? b, which means that consequent b necessarily follows from antecedent a.
(2+2=4) ? (Kamala Harris is a presidential nominee)
is false, because the consequent cannot be justified from the antecedent.
So, it is rather about a mixup in vocabulary than about intuition. I guess that many other people do that indeed too.
To be fair, if ¬(A?B) is true and A is false, anything is true.
Because, if ¬(A?B) is true, A is true.
Which isn't counter-intuitive, because it's intuitive that
A?B means not(A without B).
So it's intuitive that
¬(A?B) means A without B.
E.g. "An equation being quadratic implies it has real solutions" means not(the equation is quadratic without the equation has real solutions)
So "An equation being quadratic does not imply it has real solutions" means the equation is quadratic without the equation has real solutions.
As I alluded to in the other thread, material implication captures English usage only insofar as it guarantees that if the antecedent is true then the consequent will also be true. Similarly, the negation of a material implication says that if the antecedent is true then the consequent will be false, and this is vaguely similar to the denial of an implication in English except for the fact that the falsity of the consequent is not guaranteed in English.
The key is that in English we prescind from many things that material implication does not prescind from, such as the value of the consequent in that denial case. As another example, if an antecedent is false then the material implication is true, whereas this does not hold in English. At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not.
Quoting Deleted user
I think in examining these we are combining two confusing and non-translatable logical concepts: material implication and contradiction. Neither one translates well into English, and their combination translates especially badly.
Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).
For example, one can assert the material implication (P?Q) for three reasons:
In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this.
Exactly.
It represents an entailment A ? B, and not just a simple implication A?B.
Logic makes all its decisions by only looking at truth values while the English version assumes the existence of a system that also investigates justification.
To be fair, so does ¬(A?B).
Quoting Deleted user
Yes, because it means A without B. Isn't it intuitive that A without B entails A? And isn't it intuitive that A?B means not A without B, i.e. ¬(A ? ¬B), so that ¬(A?B) means A without B, and therefore A ? ¬B and therefore A?
Quoting Deleted user
Yep. Even if you add the irrelevant and contradictory P2, which is going to make everything true anyway.
Quoting Deleted user
Rain without wetness
[s]Wetness[/s]
Therefore rain.
Yes. So?
Logic isn't a replacement for natural languages. Nor is it a set of rules for how one ought construct arguments. This was part of the subject of my thread Logical Nihilism, and the work of Gillian Russell.
Quoting Deleted user
So what logic does in this case is to set out explicitly two ways of using "or" of which we were probably unaware. After understanding this we are able to say clearly whether we are using an exclusive or an inclusive "or". Prior to that logical analysis, we were probably unaware of the distinction, let alone which we were using.
So logic here is setting up a degree of precision that can carry over into natural languages. It's acting as a tool to make clear what it is we are doing with our sentences.
It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances. Rather logic sets out sub-games within language, with their own specific rules. Natural languages permit the breaking of the rules of any of these sub-games.
Take a look at these examples from Russell. ? ? ? and ? & ? ? ? might seem to be candidates for logical laws one might expect to have complete generality.
Identity: ? therefore ?;: a statement implies itself. But consider "this is the first time I have used this sentence in this paragraph, therefore this is the first time I have used this sentence in this paragraph"
Elimination: ? and ? implies ?; But consider "? is true only if it is part of a conjunction".
Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule.
Logic doesn't give us a crystalline replacement for natural languages. But it can set out clearly what it is we are doing with our statements.
"A does not imply B". In English that is ambiguous. It could mean:
There are instances in which A is true but B is false.
It is not the case that A entails B (same as above).
It is not the case that A implies B (where 'implies' means the material conditional).
It is not the case that A implies B (where 'implies' means a connective other than the material conditional).
Probably others.
The rest of this pertains to ordinary symbolic logic:
We have to be careful to distinguish between, on the one hand, mere implication and, on the other hand, and entailment or proof .
A -> B
is not generally equivalent with
A |= B or A |- B.
In ordinary symbolic logic, '->' does not mean 'entails' or 'proves':
A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).
A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.
A |- B iff and only if there is a derivation of B from A.
Example:
"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.
"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.
"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln".
/
Also, we need to be careful what we mean by letters such as 'A', 'B', 'P', 'Q', etc.
(In propositional logic, all formulas are sentences, but in predicate logic, some formulas are sentences and some formulas are not sentences.)
In different contexts, such letters are used to represent either:
(1) atomic formulas (atomic sentences)
or
(2) meta-variables ranging over formulas. (Sometimes logic books use Greek letters for this.)
In recent discussions, the letters are being used as meta-variables.
So, for example, when we mention 'A -> B', we understand that 'A' and 'B' range over all sentences, including ones of arbitrary complexity.
/
If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:
"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs
What do you mean by "A does not imply B"? Do you mean?:
"It is not the case that A implies B"
i.e., ~(A -> B)
which is true in any interpretation in which A is true and B is false.
or
"It is not the case that every interpretation in which A is true is an interpretation in which B is true".
Quoting Deleted user
That should be ('M' here is an interpretation):
~(A -> B) is true in M
B is true in M
therefore, A is true in M
or, if M is tacit:
~(A -> B) is true
B is true
therefore, A is true
or, without 'true':
~(A -> B)
B
therefore, A
Quoting Deleted user
'rain without wetness', 'wetness', 'rain' are not sentences.
But it does have a nice haiku-like flavor.
"If A then B" is understood differently by different people in different contexts.
So any ambiguity in "It is not the case that if A then B" stems from "If A then B".
So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B".
Quoting Deleted user
[EDIT: Dump the strikethrough potion]
[s]Arguably, they are the equivalent:
(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional)
is equivalent with:
(2) "If A then B" if and only if "There are no instances in which A is true and B is false"
So:
(4) "It is not the case that every instance in which A is true is an instance in which B is true"
is equivalent with
(3) "It is not the case that there are no instances in which A is true and B is false"
is equivalent with:
(5) "There are instances in which A is true and B is false"[/s]
Quoting Deleted user
Circling back to what? Choose whichever option you like, or add options such as relevance, or state another option.
Quoting Deleted user
Choose which option you prefer for "If A then B", then prefix it with "it is not the case that".
First, that is not idiomatic. I've never heard someone say "There is X without there is Y". Second, it could mean at least a few different things. Third, I don't know your point with the example. Fourth, the previous example at least had a nice haiku-like quality.
It's been pointed out to you at least twice that B doesn't matter:
~(A -> B) -> A
Quoting Deleted user
Then don't read it that way.
It is suggested to read it as: It is not the case that A implies B.
It is not strictly speaking a sentence, but idiomatically it is understood that it means "There is rain but there is no wetness".
"Every instance in which A is true is an instance in which B is true"
equivalent with:
"There is no instance in which A is true and B is false."
If A is false in an instance, then that is an instance in which it is not the case that A is true and B is false.
Works idiomatically. And I edited anyway for even greater sharpness:
"There is rain but there is no wetness".
is idiomatically the equivalent with:
"There is rain but no wetness".
If you disagree, then so be it.
'A -> B' is symbolic. In context of ordinary symbolic logic, it is unambiguous. What is ambiguous is everyday discourse. And, of course, many ordinary senses of "if then" don't fit 'A -> B' as 'A -> B' is used in ordinary symbolic logic. What you call an 'incongruity' stems from (1) "If then" has different sense in ordinary discourse. (2) The material conditional is not in accord with many (arguably, most) everyday senses of "if then".
Good, so we've taken care of your problem. Negation is not at issue.
Whoever first said, "if ¬(A?B) is true and B is false, A is true", the point is that it is unnecessarily cluttered by "and B is false".
For example:
Quoting Leontiskos
"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower.
Quoting Deleted user
Quoting Leontiskos
In the example I gave, "First-order claims and second-order rules of discourse."
A first order claim in propositional logic is something like, "P is true," or, "Q is false." Sentences consist of propositional affirmation, negation, and logical operators. Note, though, that, "You are contradicting yourself," or, "This is a contradiction," is a different genus, and deviates from first-order discourse, moving into the meta-language.
I thought that was the job of syntax rather than logic.
Yet if what Aristotle does in Metaphysics IV is correct, then there is a logical law that cannot be breached, namely the law of non-contradiction. Or in other words, "logic" is not a purely formal exercise. It was created for a reason and that reason has implications for reality/metaphysics.
The natural speaker assumes that there is somewhere some justification.
Formal languages may expect that too.
That is actually the main difference between classical logic and mathematical logic.
In mathematical logic, it is not just about truth tables. The goal is not limited to a bit of truth value calculus. The goal is proving entailment, i.e. (mathematical) justification.
And, that's what syntax is about. The arrangement of words and phrases in a specific order to make clear what we are doing with our sentences. Transposing them could change the meaning. So, syntax is the specific tool to make our phrases clear or let's say, 'understandable'. I don't attempt to deny the value of logic in all of this. I simply think that this is a subject of linguistics rather than logic.
Does logic make clear what we do with sentencing as Banno suggested?
I can’t see how ‘P(a)?Q(a)’ helps me to properly write: ‘the cute dog ate the bone’ for example.
Ah, I see the problem, and I carelessly extended it.
I'm dumping this:
"If A then B" if and only if "Every instance in which A is true is an instance in which B is true".
That is wrong.
As I mentioned before, there are two different notions:
(1) "If A then B"
and
(2) "A entails B"
(1) in the sense of material implication means "(A is true and B is true) or (A is false and B is true) or (A is false and B is false)". And that reduces to "A is false or B is true".
(2) means "Every instance in which A is true is an instance in which B is true".
(1) is symbolized as 'A -> B'
(2) is symbolized as A |= B
Indeed, they are not equivalent.
To which the dialetheist may simply say "so much for Aristotle".
Quoting Dialetheism, SEP
I speak English, and I don't take "if P then Q" (whether in the sense of material implication or in everyday senses, including necessity or relevance) to be about presences. Indeed, where 'P' and 'Q' are sentences, I would take "the sentence P is present", etc., to be nonsense unless it meant that the sentence P was being displayed in some way, such as on a page or screen. Indeed, I've never heard an English speaker in everyday conversation say something like "The sentence P is present". Moreover, let P stand for a sentence such as "The world is big", then I've never heard any English speaker say anything like ""The world is big" is present". Indeed, if an English said "If the world is big, then the sun is huge" then I don't know any English speaker who would say, "Yes, the presence of "The world is big" indicates the presence of "The sun is huge". Not only is that dialogue not idiomatic, but it registers as nonsense.
What do you mean by "cannot be broken"? Do you mean "cannot break without being in error"?
What does that mean?
Usually, mathematical logic is studied by means of classical logic. Indeed, mathematical logic is formulated by classical set theory. The theorems of mathematical logic, if formalized, are themselves theorems of set theory.
I agree. I just wanted to point out that syntax is a tool to make clear sense of our sentences. Not the only one, for sure. But it is one of the main tools in linguistics at least.
For example: sometimes logic formulas or axioms are not clear, but thanks to syntax we can get a better approach to understand it.
The examples I gave were:
Quoting Leontiskos
Quoting Leontiskos
So an example of a second-order rule of discourse is, "Thou shalt not contradict thyself."
I would suggest actually reading Metaphysics IV.
I am thinking of what SEP calls, "Aristotle’s Challenge to the Opponent to Signify Some One Thing."
More:
Quoting SEP | 11. Dialetheism, Paraconsistency, and Aristotle
Quoting Deleted user
"If X then Y" is incorrect because you think "If you go, then I will go" is not grammatical?
Why would an ordinary sentence form be incorrect? Every time someone says "If ___ then ___" they are incorrect?
And "If you go, then I will go" is missing a period. But otherwise it seems fine to me.
Quoting Deleted user
I don't know what you mean to say there.
Quoting Deleted user
What do you mean by "apply"?
And do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply?
What are some laws and cases you have in mind?
Quoting Deleted user
What are some of those laws of thought that can't be broken but are not laws of logic? How do you state the difference between laws of logic and laws of thought? What are the obvious reasons they can't be broken?
Quoting Deleted user
What "semantic contention"?
The most problematic foundational law in logic (Boole's "laws of thought") is in my opinion the law of the excluded middle (LEM), which implicitly assumes that the question at hand is decidable.The indiscriminate use of this law is intuitionistically objectionable:
The law of identity may also be problematic because of the existence of indiscernible numbers. However, this problem is not frequently mentioned in the literature.
The only foundational law that seems to withstand foundational scrutiny by constructive mathematics, is the law of non-contradiction:
The law is not considered unassailable, though:
They are grammatically correct in English. Why would you claim otherwise?
Quoting Deleted user
"If ____, then ___" is ordinary grammatical English.
"I am dying now" said when not dying is ordinary grammatical English, but is a false sentence.
Quoting Deleted user
(1) I know the ordinary general sense of 'apply'. But this is a particular subject, and I'm wondering whether you have an explication of your use or whether 'apply' should just be taken as undefined by you. (2) I was asking you about your use of 'apply'; I didn't assert my own use of it. I didn't assert what you quoted of me; it was part of a question to you.
And the question still stands:
Quoting TonesInDeepFreeze
But you do say:
Quoting Deleted user
What law and system are you referring to?
Quoting Deleted user
You said that there are laws of thought that can't be broken. And you said laws of logic can be broken. What are some laws of thought that can't be broken but are not laws of logic?
Quoting Deleted user
You can't conceive it. But that doesn't entail that others cannot conceive it. Also, conceiving that a contradiction holds does not entail that the contradiction holds.
Quoting Deleted user
If we put a period at the end of "If ___ then ___" , then it is a punctuated English sentence. Just as with that sentence itself.
In context of modern logic, 'decidable' means either (1) the sentence or its negation is a theorem, or (2) There is an algorithm to decide whether the sentence is a member of a given set, such as the set of sentences that are valid, or the set of sentences that are true in a given model.
LEM is not that. LEM syntactically is the theorem: P v ~P, and LEM semantically is the theorem that for a given model M, either P is true in M or P is false in M (so, either P is true in M or ~P is true in M)
Quoting Tarskian
The law of identity, the indiscernibility of identicals, and the identity of indiscernibles are different. What specific problem with the law of identity are you referring to?
Quoting Tarskian
You think that the only law that constructivism allows is non-contradiction? You've gone through all other laws and found that they are not constructivisitically acceptable?
It's not used only in logic and mathematics. In everyday discourse, people write "If ___, then" commonly. The source you cited mentioned mentions "If ___, ___" only but I would not take that to preclude also "If ___, then". Are there grammarians who explicitly disallow it? Are there not grammarians who do allow it? Perhaps there are grammarians explicitly disallow "If ___, then ___", but that would be pedantic, especially in this context, in face of the fact that "If ___, then ___" is not only used in everyday discourse, but in all kinds of writing. Moreover, since it is taken as grammatical in logic and mathematics, then that's good enough here, since logic is the subject. I don't know what point you are making about logic when you rule out "If ___, then ___".
Quoting Deleted user
As far as I can tell, it is grammatical. 'literally' is an adjective to the noun 'dying'. But the sentence is false. "I am hopelessly dying", "I am unhappily dying", "I am literally dying". Grammatical as far as I know.
Quoting Deleted user
What rule of grammar is violated. I wouldn't take using a word with an incorrect meaning is not a violation of grammar. If someone thought 'choleric' means 'melancholic', then "Jack is choleric" is still grammatical even though Jack is not choleric.
Quoting Deleted user
Yes, they are not lying or confused about their health. They simply mispoke while still grammatical.
"I am literally dying now" may be true or it may be false. But in either case, it is grammatical.
Quoting Deleted user
I would need to re-read that article, but, as I recall, dialetheism is a philosophy not a system. Though, as you mention, there are paraconsistent systems. Yes, that is an example. But, for any for any law of thought there may be a system that denies the law, so any law of thought could be denied.
If your point is that one is free to choose any system one wants to use, then, of course, one could not dispute that. But also one is free to choose whatever ways of thinking one wants to choose.
Quoting Deleted user
That something is necessary for rationality (under a given definition of 'rationality') doesn't entail that people may not break "laws of thought".
Quoting Deleted user
Whether or not I can conceive it doesn't entail that others cannot. It is not precluded that, for example, people in mystic states do experience suspension of non-contradiction. And it does not dialetheism permit conceiving such things?
You said, "Some laws of logic may express those laws of thought. But that is just a semantic contention."
Now:
Quoting Deleted user
I guess 'that' referred to the difference in the way you two stated the idea. Okay.
/
I asked, "Do you mean there are cases in which no law applies? Or do you mean that, for any law, there are cases in which that law does not apply?"
I surmise you mean the latter.
This is very close to the way that Aristotle defends the PNC in Metaphysics IV. Much of this is just a question of what we mean by 'logic'.
"the sentence or its negation is a theorem" ignores the existence of true but unprovable sentences. So, it should rather be "the sentence or its negation is true". They don't need to be provable theorems.
I do not see the difference between "the sentence or its negation is true" and "P v ~P".
Quoting TonesInDeepFreeze
I was referring to the identity of indiscernibles: ?x ?y [ ?F ( F x ? F y ) ? x = y ]
For any x and y, if x and y have all the same properties, then x is identical to y.
Quoting TonesInDeepFreeze
I was referring to Boole's laws of thought:
- the law of identity (ID)
- the law of contradiction (or non-contradiction; NC)
- the law of excluded middle (EM)
Boole did not "invent" these foundational laws but he did systematize them somewhat.
I'm just telling you what the definition is. It doesn't matter what you think "should" be or what "needs" to be.
Quoting Tarskian
The definition of 'decidable' is not "the sentence or its negation is true".
Quoting Tarskian
And that is not the law of identity. And it doesn't bear on the law of identity the way you claimed it does.
Quoting Tarskian
And constructivism uses the law of identity, so it is not the case that the only one of those three laws allowed by constructivism is non-contradiction.
That is not what I said. Straw man.
The law of identity is allowed by constructivism. It "withstands foundational scrutiny" by constructivism. No strawman.
"What it does" meaning its syntactical role, yes.
"What it means", no.
If I think 'red' means 'loud' and I say "The trombone is red", then still "The trombone is red" is grammatical even though it is false and false due to the speaker's mistake in the meaning of the word 'red'.
I didn't write '1h'.
No, 'literally' there is not violating the syntactical role of an adjective.
And at this point, you are merely arguing by reiteration of your assertion.
"Bob has a red French horn" is syntactical even though the speaker meant that Bob's French horn is loud.
Yet you cited one.
Quoting Deleted user
I thought you might have intended some point about logic. Good to know that you didn't.
All three laws are allowed. I just pointed out that there are issues in assuming two of them.
What does 'speak wrongly' mean? Speak ungrammatically or speak falsely?
Of course it is possible to use the wrong word and still be grammatical. People do it all the time.
One could make up examples all day, or observe them.
'literally' is an adjective. "I was literally dying" is grammatical. It is not made ungrammatical by the fact in the world that the speaker happened to not be dying and not literally dying.
The way it read was that there are laws of logic that may be broken but not laws of thought. But if any law of logic may be also a law of thought, then there are laws of thought that may be broken too. And it wasn't stated as to what systems may deny, but merely as to what laws may deny.
Grammar doesn't dictate what is true or false, only what is well formed.
"I was literally dying" is well formed even if untrue.
"Bob's French horn is red" is well formed even if untrue.
We can give millions of examples in which the speaker misuses a word, but the sentence is still grammatical. Since you are wont to skip that point, here's one more:
The speaker may think 'melancholic' means 'mellow', then say, "The song is melancholic" when the song is not at all melancholic. A false but grammatical utterance.
Are you going to continue to skip that fact?
No, you said that the only law that "withstands scrutiny" for constructivism is non-contradiction. And that is false.
What is regarded as rational may be different for different people. And people may choose even to think irrationally by any standard. But, of course, given a particular conception of rationality, some thoughts will not be rational and will violate certain attendant laws of rational thinking.
Quoting Deleted user
Doesn't matter what the definition is. People may break all kinds of norms of rationality in their thinking. But, of course, tautologically, they can't break those norms with out breaking those norms.
Quoting Deleted user
Whatever one thinks about dialetheism, the point stands that people may conceive dialetheistically. A person may say of himself that he cannot conceive other than by certain rules regarded as irrational not to conceive by. But that doesn't entail that other people can't conceive outside of those rules. Indeed, in such things as art, dreams, ruminations and mystical experiences, people can conceive in all kinds of ways. But, again, if the point is that people can't think irrationally without thinking irrationally, then of course, it would be irrational to deny that point.
Agreed. The identity of indiscernibles is criticized in other areas of mathematics.
Regarding constructivism, we were talking about the law of identity.
What are some criticisms in mathematics of the identity of indiscernibles? (Of course, it is not first order axiomatizable.)
Thank you for that cite. That's interesting.
By syntactical, I mean grammatical.
"Bob has a red French horn" is grammatical, even though it is false and even though Bob is misusing the word 'red' when he means 'loud'.
But not ungrammatically.
"My cat is black" is grammatical even though it is false and the speaker meant that his dog is white.
You keep evading that very simple point.
Ah, how conveniently you left out 'semantically'.
It is wrong semantically, as it uses the wrong meanings of the words. It is semantically wrong, but not grammatically wrong.
You keep evading:
"Bob is a splenetic guy" is grammatical even though the speaker misused the word 'splenetic' thinking it means what we mean by 'splendiferous'.
As to teaching English, of course it is needed not only to say that the sentence is false but that it is false because the words don't mean what the speaker thinks they mean. But that still doesn't make "My cat is black" ungrammatical. It is both (1) False and (2) False on account of the wrong words being used. But it is still grammatical.
When we consider whether an utterance is grammatical, we don't first check what the speaker meant by the utterance. We merely look at the words themselves. If I give you this:
"The cat is black" and ask, "is that grammatical?" You don't track down the speaker and find out whether he knows the definitions of 'cat' and 'black'.
My mistake about 'adjective'; I do know that it is an adverb.
But there's another example:
"'literally' is an adjective" is grammatical, even though false, and even though it is false by dint of the speaker using a word incorrectly.
And I'm merely talking about the fact certain parts of speech are required to be certain positions and in relation with other parts of speech. "Black the is cat beautiful' is not grammatical as the part of speech are not in correct order, but "The black cat is beautiful" is grammatical.
It's interesting that you say that. Because it is very wrong.
Semantics concerns the meanings of words. Syntax (grammer) concerns the rules for formation of expressions.
If people have different concepts of rationality, then they may differ as to what laws of thought they adhere to, thus there are laws of thought that may be broken.
Quoting Deleted user
That is exactly what I am not saying. I am not at all saying that rationality permits irrationality. Rather, I am saying that people may break rationality, thus they may break a given law of thought. I even said that, of course, tautologically, adhering to rationality requires adhering to rationality.
Quoting Deleted user
That's okay. But it is different from saying that the laws of thought cannot be broken. If we consider those laws of thought to be necessary for rationality, then they cannot be broken without incurring irrationality. But they still can be broken.
Quoting Deleted user
But, if I recall correctly, you said that in general laws of logic can be broken, as you even gave an example of breaking the law of noncontradiction. Moreover, if there is a single law of logic that can be broken, and that law of logic corresponds with a law of thought, then there is a law of thought that can be broken. Moreover, even that point is not required, since we know that people do break laws of thought. Though, of course, if a certain law of thought is required for rationality then it can't be broken without incurring irrationality.
Syntax and grammar are synonymous in some contexts and nearly synonymous in others.
Semantics stands opposed to them.
Look it up.
Quoting Deleted user
When you add emphases (such as bold or italics) to my quotes, you should indicate that the emphases were added.
Right, my typo.
Now, you're arguing by reiteration of your claim. When it comes full circle like that more than once, rational discussion is diminished.
What? You don't know how "[emphasis added]" works?
My original did not have bold. You added bold to my quote. When you do that, you should include a note that you added the emphasis. It is not up to the reader nor me to recall the peculiarities of the formatting processes of this site to then reason, "So the bold would have been lost if it were quoted, so if it appears, then it must have been added."
You just need to put in "[emphasis added"].
Whether it was three decades ago or three seconds ago, it is not proper to display someone's quotes with emphases they didn't use unless you indicate that the emphases were added.
grammar
: the study of the classes of words, their inflections (see inflection sense 2), and their functions and relations in the sentence
: a study of what is to be preferred and what avoided in inflection (see inflection sense 2) and syntax (see syntax sense 1)
: the characteristic system of inflections (see inflection sense 2) and syntax of a language
: a system of rules that defines the grammatical structure of a language
syntax
: the way in which linguistic elements (such as words) are put together to form constituents (such as phrases or clauses)
: the part of grammar dealing with this
/
Nothing there about semantics or meanings. Rather, the structural aspects.
Especially in logic and philosophy of language, usually 'syntax' and 'grammar' are understood together. And semantics is different. Syntax concerns whether an expression is well formed. Semantics concerns the meaning of the expression.
But one of yours [emphases added]:
Quoting Deleted user
Not that I trust "Google dictionary", but you proffered it. So:
The usual sense of 'grammar' is 'syntax'. But sometimes it includes semantics. So I will award myself the point that usually 'grammar' and 'syntax' are used the same. I will award you the point that sometimes semantics is included. But consider that in logic, usually a sharp distinction is made between syntax and semantics and use of 'grammar' would align with 'syntax' not 'semantics'. The quotes you give do indicate a more extended sense of 'grammar'. I haven't seen that sense in logic or philosophy of language, but if you insist. Meanwhile, you could have easily ascertained that 'syntax' and 'grammar' are commonly used interchangeably but that your context is different.
Oh wait, the Google entry is just the Oxford entry, so as you posted it redundantly, I will too:
Quoting Deleted user
I'm not talking about guessing what post was quoted. I'm talking about the fact that it is ridiculous to expect a reader to factor in the peculiarities of the formatting of quotes to know whether the emphasis was original or added.
A reader shouldn't have to click back to find out where the bolding came from. It is the responsibility of the quoter - not the reader and not the quotee - to indicate that the emphases were added.
You're reiterating your claim as you made it at the start of this round.
So, I'll reiterate:
"Jack is happy" is grammatical even when the speaker misused the word 'happy' while thinking it meant 'doleful'.
We don't have to ask the speaker what he meant to check whether he knows the correct meanings of the words. We just have to look at the sentence to see that it obeys the formation rules for the language.
And, in logic, which I hope was the original context, the usual distinction is between syntax and semantics, and with 'grammar' sometimes mentioned rather than 'syntax'.
Now we're full circle more than three times at least.
He doesn't.
No, we don't need to make any such assumption. You're just stipulating that out of thin air.
We might know nothing about who or what wrote an expression.
If I display a sentence on a piece of paper and leave it on the sidewalk, and you pick it up and read it, "The cat is black", then you recognize that as grammatical, no matter whether written by Shakespeare or a random word generating machine or an insane person who thinks 'cat' means 'screwdriver' and 'black' means 'wet'.
Suppose you have a job correcting school assignments, and you are never in the classroom, never met the kids, you just correct the papers. Then you don't know the vocabularies of the kids. You don't know which ones know the correct definitions of the words used. But you can still correct and grade the grammar. If you see, "The car engine is noisome", then you mark the sentence as grammatical, even though you don't know whether the kid knows that 'noisome' means 'noxious' and not 'noisy'.
Really, that is so plain that if you still refuse to understand it, then indeed you defy rationality.
It's a nod to Saussure (and his demonic, hyper-nominalist post-modern semiotics of destruction—as opposed to our Augustine and Co.'s virtuous and sure triadic semiotics of life).
English-speaking philosophers who embrace contemporary French philosophy have to be very careful to use the French, lest they commit a massive faux pas like pronouncing Derrida's "différance" with a French accent. This is embarrassing indeed. The whole point is that différance and différence are pronounced exactly the same, and so one can only tell the difference when looking at them on paper (the victory of the logocentric over the phonocentrism of Saussure's Grammar, which has now been deconstructed).
However, the idea that signs might have something to do with a res or referent is terribly naive, if not downright provincial. "Intent of the speaker?" In grammar?
Friend, surely you know that both the author and the utterer have been dead for decades now? It would be totalitarian, not to mention brutish to suppose that intent should be allowed to tell people how they should construct their meaning, or that the properties of res/signified should play any determinant role in shaping the meaning/dicible in some sign relation. After all, people are themselves just signifiers, nexuses of sign-based discourses.
The syntax is not fine. (1) 'have' should be 'has' (2) 'piink' is not a word (3) 'forhead' is not a word'.
Syntax checks. Are these words? Are the words in an allowed order based on their kind? Are the words in correct case, inflection, etc.
Semantics checks: What are the meanings of the words? What are meanings of the clauses? What is the meaning of the sentence?
Same with formal logic. Syntax cheks: Are these symbols of the language? Do the sequences of symgols form formulas? Are the sequences of formulas allowed as proofs according to the rules?
Semantics with formal logic: To what do the symbols refer? What are the truth values of the sentences based on the meanings of the symbols and subformulas?
I can't believe you stooped to such a sophomoric argument. Obviously, we consider a context in which we at least agree as to the kind of word. Your argument is horrible desperation.
And still it doesn't answer that the "The cat is black" is seen to be grammatical even if the author of the sentence is anonymous, and even if we don't know whether the author is a human being or understands anything about language, as the expression could have been randomly generated and only by luck came out grammatical.
For about the dozenth time:
If you show me "The cat is black" then I will mark it as grammatical, not matter where you got the sentence.
Which brings me back to my point that is sustained:
Quoting TonesInDeepFreeze
People operate mentally in all kinds of ways: Fictionally, absurdly, poetically, ironically, day dreaming, dreaming, mystically and insanely. But your point reduces to the tautological: the mind can't operate rationally without operating rationally. No one disagrees with that.
And so we've come around again full circle.
That is not at issue. What is at issue is whether that law of thought can be broken. Yes it can. Of course, if we hold that it is required for rationality, then we may say it can't be broken rationally. But that doesn't refute that people break laws of thought often. Full circle again.
I have found Merriam to be good, especially unabridged, but some deterioration over the years. I read all of yours. I mentioned the others for emphasis.
Quoting Deleted user
From definitions you posted yourself.
/
If I were to nitpick, it would be a whole other thing. Being careful to state things about logic accurately so that false conclusions about are not drawn is not nitpicking.
/
Morphology concerns form. And so also syntax. Especially in logic, syntax is a mater of form, hence 'well formed'.
Quoting Deleted user
In logic, terms are formed by rules. If symbols are not in correct order or incorrectly omitted, then they are not syntactical. In that way too, if a string of letters doesn't even form a word, then the expression in which the string occurs cannot be syntactical.
Quoting Deleted user
I didn't say 'morphological cases'.
Your argument stooped to the tactic of citing ambiguity as if we would not be discussing modulo certain ambiguities.
But you are. Right now. Anyway, my posting is not based on whether you read or don't read.
I explained it when I first flagged you on it.
Oh, please! Talk about inane nitpicking that isn't even correct! Obviously I'm using 'grammatical' in the sense of 'conforming to the rules of grammar'.
I responded exactly regarding the post of mine that you referred to.
You said that I nitpick. I don't. But then you turn around and incorrectly nitpick!
I don't know your point. Anyway, people may use the word 'logical' differently: (1) pertaining to logic or (2) logically correct.
And so, your incorrect nitpick about my use of 'grammatical' when obviously I mean 'grammatically correct' or 'according to the rules of grammar'.
One instance that I can see might be regarded as nitpicking was when I said saying "B is true" was extraneous. But I mentioned it in a stylistic sense that it's better not to include extraneous items so that the arguments can be seen more clearly, without the distraction of those items.
Are you saying the poster's sentence is not adequate English?
Then I overlooked that it did.
Right, I was just pointing out that this is almost certainly what the reference to "in modern theory," was referring to. This is how the French makes it into English sources.
Saussure is relevant to the conversation at hand in that his later post-structuralist disciples eventually worked themselves towards totally divorcing meaning from authorial intent and context. And this move was given an almost political connotation, a "freeing of the sign." Although one might question if some of the further evolutions of this way of thinking might not just succeed in freeing language from coherence and content.
I think there is actually a connection here to how formal grammar is conceptualized. In either case, the focus becomes signs' relations to other signs, pretty much to the exclusion of context or content.
Yes, it doesn't follow. No one said otherwise. And yes, I was referring to your notion of the laws of thought. I'll say it again:
One can break the laws of thought on pain of being irrational. But you say that the laws of thought are unbreakable. But one can break the laws of thought. So you regroup by saying that one can't break them and be rational. But that is not at issue. My point is that one can break the laws of thought, contrary to your earlier claim.
Quoting Deleted user
Definition of what? Of 'the laws of thought'? Repeat or not repeat whatever you like.
Quoting Deleted user
Where is there a report that modern writers in general believe that laws of logic may not be understood as expressing laws of thought? And what period do you regard as modern?
Quoting Deleted user
I'm uncertain whether I understand you. Certain systems don't respect certain laws of thought. That doesn't entail that laws of thought cannot be broken. Indeed, it evidences that they can.
Also, you say "the latter", which is 'laws of logic'. So 'they' also refers to 'laws of logic'. And you say 'they may only allude to or be based on laws of thought'. So that is saying that laws of logic may only allude to or be based on laws of thought. But that seems the opposite of anything we've agreed on. If the laws of thought require rejecting contradiction, then systems that allow contradiction do not adhere to that law of thought.
Quoting Deleted user
(1) In one case, I was unclear as to whether you were quoting in agreement with the quoted poster. And I overlooked that your recent lashing out was not directed at me. That is not even remotely constant (2) In this instance, I've been in exactly the right place about what was posts was referenced.
Quoting Deleted user
Your replies don't even come close to a refutation.
It's plain as day: One can easily see that "The cat is black" is grammatical, without having to know anything about the person who said it, or even if it was not said by a person but formed randomly by a machine. You've not refuted that. One of your replies is that we assume the speaker knows the meanings of the words. But that is not necessary to see that the sentence is grammatical. We could say, "I have no idea whether the person who wrote "The car engine is noisome" knows that 'noisome' means 'offensive' not 'noisy' but that doesn't matter if all you want to know is whether the sentence is grammatical. I'll happily and without any reservation tell you that is."
/
Oh, and about nitpicking: Your objection to "If ___, then ___" is a doozy!
You just completely ignore the point, that I've made twice, now a third time:
In such mental states, people often break the laws of thought.
I'm happy to read any definition you'd restate.
The refuted person may not be disposed to accept that he's been refuted. But it doesn't follow that if a person points out that he's not been refuted (and gives clear argument about that), then that person is doing that because he doesn't want to admit to having been refuted.
I should list "prerequisites" for talking about logic.
And a fact about minds is that they are often irrational.
Actually, easier just to list a three book course, which I've done several times in this forum.
I thought they were two different definitions. But the second includes additional assertions beyond what I would have thought is a definition. Also, I don't know what 'instead' refers to.
Whatever is "subjacent", in those mentioned mental states, the laws of thought are broken in the sense of irrational thinking, believing or imagining. If a mystic experiences contradictions as being true, then he's not breaking the laws of thought? If one dreams that one's great-grandfather is both alive and dead at the same time, one is not breaking the laws of thought?
Laws of thought are facts about your mind such that those facts are necessary for the operation of the mind.
I don't know if that's what you mean, but it's my best guess.
Or maybe just say:
Laws of thought are the necessary mental conditions for the operation of the mind.
From that definition, it follows that they can't be broken.
/
So, when a person is utterly irrational, they are still obeying the laws of thought on account of the fact that there are mental conditions necessary for the operation of their mind?
I didn't mention you skills. I mentioned your knowledge.
And you don't have to feel they that my view is needed nor do you have to request it for me to state it.
Meanwhile, you lashed out at another with your characterization of his knowledge of language. Same applies to you in your knowledge of logic. You've made hundreds and hundreds of posts about logic that are a dead end as your gravamen can be neatly summarized in a couple of sentences (as I did for you) without the pointless variations all on the same pointless theme.
I don't require your courtesy. And I don't require you not to post so that you don't wear out my patience as you do. Anyway, in general, many people in this forum will be discourteous quite soon after they are disagreed with.
[EDIT: "courtesy" from a guy who makes a ridiculous argument against the common courtesy of noting that emphases were added to a quote.]
You can post or not post as you please. And I'll do the same.
I don't pretend to be a bully and I'm not one. And "senile" is to guffaw.
Meanwhile, no matter how you regard me as "coming off", I don't manufacture perceptions about you in that way. No matter how you "come off" to me, I regard the substance of your posts, good or bad, on their own terms, not personally.
Of course they're possible. Whether in absurdist day dreaming, insanity, dreaming or in mystic state, one can have all kinds of irrational thoughts and dispositions. Even in everyday experience, people often drift to sleep with disconnected nonsensical ideas and irrationality.
Quoting Deleted user
Yes, and therefore "laws of thought" pretty much reduces to simply "conditions necessary for mentation". If whatever one thinks, no matter how irrational, is not breaking the laws of thought, then the notion of 'laws of thought' is so general that it is hardly worth mentioning. That suggests putting some more meat on the bones of your definition.
Synesthesia does occur. And people have all kinds of false beliefs not derived by good inferences. But beyond those, people also have even more profoundly alternative states.
Quoting TonesInDeepFreeze
I should not have honored that garbage even by laughing at it.
"senile" is juvenile. Worse, it's pernicious. One would think that such crude ageism wouldn't get into public past the lips of a putatively aware poster. People have mental difficulties for many different reasons. It's not a matter of age, but of the difficulties no matter their cause. Meanwhile, bigoted ridicule of people for their age is obnoxious and disgusting. Also pretty bad is to compound that bigotry by making it a term of general insult against targets whose age is not even known and not relevant no matter what it is.
:smirk:
The oddity is that there is not parity between a conditional and its negation:
Quoting bongo fury
A conditional, by its very name, signifies that which is not necessary. (1) is therefore conditional in that it neither commits us to A, ¬A, B, or ¬B. It retains something of the hypothetical nature of natural-language conditionals.
(2) is not a conditional in this sense, for it commits us to both A and ¬B.
In natural language when we deny a conditional we at the same time assert an opposed conditional; we do not make non-conditional assertions. In natural language the denial of a conditional is itself a conditional. But in propositional logic the denial of a conditional is a non-conditional.
See:
Quoting Leontiskos
Quoting Leontiskos
And:
Quoting Leontiskos
"If lizards were purple then they would be smarter."
The denial is, "Even if lizards were purple, they would not be smarter." It is not, "Lizards are purple and they are not smarter." The logical negation is the English counterexample.
---
The deeper issue here is that there is no uncontroversial way to translate between English and formal logic, because English has inherent meaning where logic has none. Logical meaning is derived from English meaning, and not vice versa. Because of this the "meaning" of a logical sentence is merely what can be done with it, or what it can be transformed into, and no one transformation is more central to its "meaning" than any other. This is what I was trying to get at on the first page.
Bongo did a good job of using English to capture the range of the logical possibilities, but at least one problem arises in that the English negation and the logical negation are substantially different. As you pointed out in the other thread, a central aspect of an English negation of a conditional is that the consequent and only the consequent is negated (e.g. If <lizards were purple> then NOT<lizards would be smarter>).
(What this then means is that to unequivocally claim that ' scenario does not represent a contradiction is to rely exclusively on a "bug" of material implication, and only those who are able to contextualize material implication within a larger whole will be able to consider the question more fully.)
Is there something wrong with: (A?¬B)?
(This is why I added a parenthetical edit to my last post, which is about the OP of the other thread.)
Quoting Deleted user
Yes, for it is not possible to capture the negation of the idiosyncrasies of material implication while simultaneously capturing the negation of the notion of implication or conditionality. One or the other must be lost. English abandons the first and propositional logic the second.
If B is always false whenever A is true, then surely «A does not imply B». The logic covers the English but the English is not captured by the logic.
((A) does not imply B) is quite different to (A implies (not B)) I would think.
P: Lizards are purple
S: Lizards are smarter
I think the English sense is never falsified by the logical sense, and in that way it would seem to hold. The problem is that the logical sense can be falsified by alternative English senses, given that English has a more robust notion of implication than material implication. So you can't go in the other direction. Ergo, you cannot translate (P?¬S) as, "If lizards are purple, then they are (necessarily) not smarter," even though you can draw the conclusion, "P does not imply S" ("Lizards' purpleness does not imply lizards' smartness").
Edit: So we might say that (1) guarantees (2) but (2) does not guarantee (1). Thus I admit that it doesn't count as a real translation.
Edit2: I think Janus' argument is special insofar as it makes use of a Cambridge property, and in that case (1) and (2) seem to be the same.
It's not the case that A implies B:
~(A?B)
[s]But ¬(A?B), ¬B |= A is invalid.[/s] And it's not one of De Morgan's rules, which are equivalences between "^" and "v".
Oops.
So then why is it that the logic cannot capture the English, "A does not imply B"? Is it because the English represents a denial without any corresponding affirmation?
If so, it seems that I was wrong when I said that to deny a conditional in English is necessarily also to affirm an opposed conditional:
Quoting Leontiskos
In English we can deny in a manner that does not affirm the negation of any proposition, and this violates the way that propositional logic conceives of the LEM. In fact, going back to flannel’s thread, this shows that a contradiction in English need not take the form (A ^ ~A). In English one can contradict or deny A without affirming ~A.
...but then again maybe to say “Not A and not ~A” is only open to Buddhist-type logic or Buddhist-type English. Even if that is so, perhaps what is available more broadly is the denial of a consequence without any attendant affirmation, such as, “That does not follow from this, and I make no claim about what does follow,” as I claimed <here>. In this way one undercuts an inference and deprives the conclusion (or consequence) of its validity without falsifying the conclusion. Thus one can say, “A does not imply B,” without making any positive assertion, conditional or otherwise. Apparently the relation between a negation and an affirmation differs in English and in logic.
Edit: This may actually be key to understanding A?(B?¬B), for the contradiction is nonsensical or unstable when taken in a particular sense, and this is why the standard logical operations cannot be applied to it in the same way. A reductio ad absurdum may be parallel to the English move of denying a conditional without affirming anything in the same move. If a reductio affirmed something in the same move then there would be no and-elimination step, and if that were so then a reductio would be identical to a modus tollens, which it is not. The affirmation involved as the final step of a reductio only takes place "after" the and-elimination step. The contradiction is repugnant regardless of which conjunct is preferred (or of which supposition was originally made), and this makes sense because what is proximately aimed at in a reductio is contradiction per se—a universal concept—rather than the application of any truth value to a variable. The application of the truth value to the variable is what is remotely aimed at, and will only take place after the contradiction and the and-elimination have already occurred.
Couldn't it be said that logical language establishes a number of precise connections between states, but the absence of a connection is not defined. It's the negative space that remains outside the ruleset.
Quoting bongo fury
doesn't follow from
Quoting bongo fury
then it would seem that we don't intuit negation in this case as a photographic negative of the Venn diagram, which is what logic would deliver. In which, i.e., all previous no-go (shaded) areas are declared open for business, and all previous open regions are shaded out. Rather, the intuition is that a (in this case the) previous no-go area is opened up. But nothing closed off. We wish to withdraw or deny an assertion without thereby committing to its negative. Deny it is the case there won't be a sea battle, without claiming there will.
So, not really negation. Not cancelling out the first. Not restoring not(A without B) to A without B.
¬(A?B) appears to suggest, intuitively: maybe A without B, maybe not. No commitment. No information. Tautology. No shading in the Venn diagram. (Whose 4 non-overlapping areas correspond to A & B, A & ¬B, ¬A & B, ¬A & ¬B.)
Leaving it open.
If.
Yep, I think this is right, and it's what I was trying to get at on the first page. I think my point about "denying without affirming a propositional negation" is also right, and Bongo developed that point. I wonder if the two can be brought together.
Quoting Deleted user
In general I want to avoid thinking any English represents the logic, but I also I think this is a good point. But to give something of a counterexample, if A is false then we can say A?B, and yet your English does not capture this move. Thus:
Quoting Leontiskos
Keeping to this counterexample, "not A without B" captures a truth-functional conditional, but it does not fully capture a material conditional. English involves "causation," but it additionally prescinds from the idiosyncrasies of the material conditional. I think Bongo's negation may have more to do with the materiality of the conditional than its lack of causation, although the two may well be related.
Quoting Deleted user
Ha - that's an additional consideration that I was not thinking of (Diotima's point in the Symposium). Prescinding from this question and from the question of Buddhist logic, my point is primarily about conditionals or consequences, and can be set out in response to Bongo:
Quoting bongo fury
Basically, but more precisely, I would say that we are denying an inference. In English we don't usually say, "You are wrong that there will be a sea battle tomorrow, and yet there may be a sea battle tomorrow."* Instead we say, "Your reasoning for why there will be a sea battle tomorrow is not sound, and yet there may be a sea battle tomorrow."
N: There is a wind from the north tonight
S: There will be a sea battle tomorrow
N?S
The denial in English would seem to be, "S does not follow from N." This doesn't mean that S will always be false whenever N is true. It only means that S need not be true when N is true. This seems to be evidence for Lionino's view that a causal connection is at play. Or as I said on the first page, "The English has to do with a relation between P and Q that transcends their discrete truth values."
Quoting bongo fury
These are good thoughts, but I think a kind of cancelling-out is taking place. It's just that the denial transcends the limitations of truth-functional logic.
To deny something requires understanding what is first being asserted, that it might be denied. If someone says, "Wet grass follows from rain," they are not asserting everything that is involved in the logical claim <[rain]?[wet grass]>, for they are not asserting the idiosyncrasies of the material conditional, such as the idea that ~[rain] justifies their claim. At the same time, they are asserting something more than the logical claim insofar as they do not believe that the falsity of their claim would mean that rain always produces dry grass. Something more subtle is being said and something more subtle is then in turn being denied, and these subtle affirmations and denials don't straightforwardly translate into the affirmations and denials of classical propositional logic.
Or going back to my earlier post and putting it in simpler terms, we can deny a conditional with a simple denial of the metaphysical relation, or else with a counter-conditional, or else with a counterexample. When classical propositional logic denies a conditional it is limited to doing so with a counterexample (e.g. N ^ ~S). This is something of a bug, for to deny the essence of a conditional is to deny its conditionality (e.g. "N does not ensure S"). English is capable of all three responses; propositional logic is only capable of one.
*I am changing the proposition to avoid confusing double-negatives.
Yes, the red and white system at least. Unfortunate that it shades in where I was shading out. But it shows how logic uses "not" as a reversal of shading, sending anything in row 2 to row 4 (and vice versa, and also reversing shading within row 3). Whereas ordinary language, while it might do that, might equally well signal a retreat to the very top, leaving all options on the table.
Or (@Leontiskos) it might do something else more elaborate which deserves analysis. Rabbit holes galore, of course. :grin:
So I think you are overstating this idea. Conditionals have a directionality that partially mimics causality. Meta-logically, they are intended to support the inferences of modus ponens and modus tollens. These are directional, asymmetric inferences. When you think in terms of coexistence or when @bongo fury speaks about "Not A without B" or Venn diagrams, you both seem to be thinking primarily in symmetric, non-directional terms. For example, "coexistence" is not asymmetrical or directional like (A?B). It is symmetrical like (A^B).
Further, "A implies B" does not necessarily mean that A causes B. As I said on the first page, the relation can be indicative. For example, the antecedent can be a sign of the cause that is the consequent. "Warmth implies fire," does not mean that warmth causes fire. "Implies" can also be correlative, where two correlates are caused by a third thing, but this is a true case of coexistence, in which the relation is biconditional (and therefore symmetric) rather than merely conditional (and therefore asymmetric).
There are lots of legitimate ways to speak about (A?B) in English, and each is incomplete:
[*]Forms relating to modus ponens:
[*]Forms relating to modus tollens:
[*](I omit the forms relating to the idiosyncrasies of material implication)
[/list]
There are also lots of legitimate ways to speak about ¬(A?B) in English, and each is incomplete:
[*]Forms relating to the denial of modus ponens:
[*]Forms relating to the denial of modus tollens:
[*](I omit the forms relating to the idiosyncrasies of material implication)
[/list]
Of course some of these overlap. For example, the multiple meanings of "without" make "Not A without B" ambiguous between a directional modus ponens and a non-directional ¬A?B.
Again, "No one transformation is more central to [the logical] 'meaning' than any other" (). Privileged meanings only emerge at the meta-logical level:
Quoting Deleted user
These are equivalent at the first-order level, but not at the meta-logical level. At the first-order level they are both true and there is no difference between the truth of (1) and the truth of (2). At the meta-logical level, (1) partakes in the true purpose of a conditional whereas (2) does not (link). (2) is a consequence of the idiosyncrasies of the material conditional. This relates to my earlier point that the logical negation of a conditional is no longer a conditional, and in that case the modus ponens and modus tollens are no longer accessible, and because of this the directionality of the conditional dissipates.
If one does not make the meta-logical distinction between (1) and (2) then they will be tempted to claim that conditional logic cannot map asymmetrical or directional relations (including causation). This isn't right. A conditional can map an asymmetrical relation. Can it map something like causality? Yes and no: partially but not fully, because causation is not entirely truth-functional.
The key here is that propositional logic distinguishes (1) from (2) not in themselves, but extrinsically through modus ponens and modus tollens. Even though (2) is 'true', nevertheless it cannot be used to draw any substantial conclusion. Calling the conditional "true" is just a useful fiction which has no practical impact on the system. Or rather, it shouldn't. In the other thread we are seeing the havoc that meta-logical ignorance can wreak, for to permit standing contradictions gives the "dross" of the material conditional a potency it was never intended to have. It turns the useful fiction into non-fiction.
I suppose it is worth asking whether these are the same two inferences, and whether the first is any more "directional" than the second:
I want to say that they are different mental inferences, which is why we think of them differently (in English). But this is part of the difficulty of the thread. "Rabbit holes."
One is an inference of B from {A -> B, A}.
The other is an inference of B from {~A v B, A}
However, A -> B and ~A v B are equivalent, so the inferences are different but equivalent.
As to "directional", we'd need a definition of "directional".
What "rabbit hole" there is depends on the silly rabbit looking for real or imagined rabbit holes.
I think you may have mixed up a bit of the verbiage there, but I think you are saying that «not A without B» prescinds from whether or not ¬A justifies the conditional, and that is precisely my point. «not A without B» does not capture the fact that ¬A makes the conditional to be automatically true.
Or in other words, I can say, "¬A, therefore A?B," and clearly «not A without B» does not justify such a move. If all we knew about A?B was «not A without B», then we would not know that such a move is valid.
Okay, so we have:
What I am saying is that knowledge of (2) does not give us knowledge of (1), and yet everyone who knows what A?B means has knowledge of (1). Therefore (2) does not give us complete knowledge of A?B. (2) does not fully represent A?B.
(Edit: I am pointing to a problem with your claim that we can translate A?B into English as "Not A without B.")
Quoting Deleted user
No, your conclusion (A is true) is not valid. You seem to be interpreting “¬(A?B)” as: “¬A->¬B”, and that’s invalid. “¬(A?B)” just means that the truth value of A does not give us a clue as to the truth value of B. A better English translation of ¬(A?B) is : it is not the case that A implies B
Consider these substitutions:
A=All bluebirds fly
B=Fred is a duck
This is consistent with ¬(A?B) being true. If we discover Fred is a pigeon then B is false, but it tells us nothing about whether or not all bluebirds fly.
I think this is simply incorrect.
Quoting Deleted user
Again:
Quoting Leontiskos
<"Not A without B" does not preclude ¬A> is a different proposition than <If ¬A is true, "Not A without B" is true>.
(1) is true. (2) is false. It is false for you to claim that the consistency of ¬A and "Not A without B" justifies (2). (2) requires more than consistency. It requires more than that ¬A is allowed. "¬A is allowed, therefore (2) is true," is an invalid claim.
Put differently, we can know from «not A without B» that ¬A is not disallowed, but we cannot know that the statement is made true by ¬A.
For something of a disambiguation, see:
Quoting Leontiskos
I concede your point, but what you have proven is that:
¬(A?B)
Implies A
(Which I confess seems counterintuitive - see below*).
You had said: If A does not imply B, and B is false, A is true
That second premise(¬B) is superfluous to the conclusion (A).
--------------------------------------
*Now suppose we apply the logic to these statements:
A=All bluebirds fly
B=Fred is a duck
¬(A?B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")... which is certainly true because the antecedent has no bearing at all on the consequent
(¬A?B) = "not all bluebirds fly" or "Fred is a duck"
...
A?¬B: All bluebirds fly and Fred is not a duck
Problems:
Despite the fact that ¬(A?B) is a true statement...
1) it is NOT true that all birdirds fly (hatchlings don't fly),
2) My pet duck is actually named Fred.
But the logic conclusion says otherwise.
Something ain't right. I had to dig out my 1973 Logic textbook to understand the problem, but I'd like to see if anyone can identify the problem on their own.
The "without" reading of A?B does need brackets when written:
Not (A without B)
i.e. ¬(A & ¬B)
I think they are there implicitly in "not A without B" as spoken. So the spoken phrase does clarify the logic of ?.
But perhaps they are needed explicitly when the phrase is written. I mean,
(not A) without B
seems a willful misunderstanding. And gives (¬A) & (¬B).
But brackets will prevent that particular misunderstanding.
Ah, okay, I see where you are coming from now. It seems like a strange interpretation:
In English it is usually different to say, "Not A without B," and, "Anything which is not A without B is true."
Moreover, A?B does not follow from Ephraim or Gregory's answers in the way that «not A without B» does, and Daniel's answer seems to falsify «not A without B» without falsifying A?B.
Quoting Deleted user
I was thinking of ¬¬(A?B)?¬(A?¬B). This is not the same as your interpretation of "Not A without B."
Quoting bongo fury
This is why I would prefer "No A without B." The "parentheses" (however one wishes to depict them) become more important when you want to transform the proposition logically, or draw a modus tollens, etc.
Sure, and in English is to say A without saying anything about the value of B to say A without B? It would seem so.
Would anyone interpret "Not A without B" as A?B unless they knew ahead of time that they were supposed to interpret it that way? It seems highly doubtful.
Quoting Deleted user
The technical problem here is that the English "Not A without B" in no way circumscribes the domain as ((¬)A, (¬)B) pairs. Neither Benjamin, Caleb, Ephraim, or Gregory are saying A without B, and yet only Benjamin and Caleb's answers entail A?B.
For example, the only way to claim that Gregory's answer does not entail "Not A without B" while Benjamin's does, is to beg the question and assume that "Not A without B" is equivalent to A?B. Without that assumption there is no reason to think it is correct that ¬A ? «Not A without B» and incorrect that C ? «Not A without B».
The statement "It is not the case that ("all bluebirds fly" implies "Fred is a duck") IS true. But you're right that it's not equivalent to :
¬(A?B)
But why isn't it? It's because there is no material implication. The formula (A?B) cannot be used in all semantic instances of "if A then B".
I don't think I ever realized this before. When I took sophomore logic (50 years ago!), we concentrated on formulaic proofs. But the mapping to semantics is critical.
Actually, yes, I think they would. People tend to understand that arrows signify directionality, in the sense of starting point ? destination.
Quoting Deleted user
Sure: 2% of people might interpret it as, "No A without B," but that doesn't make for a very good translation.
Quoting Deleted user
It is supposed to be simple: Has Benjamin agreed with Aaron? Has Caleb? Has Daniel?...
Or else: Aaron gives the condition, "Not A without B." Have the others fulfilled that condition or failed to fulfill it? The most obvious fulfillment would be, "A with B."
My point is that even the 2% who interpret it as, "No A without B," don't quite know what they mean by that. The real translation in those terms is something like, "No A without B in the domain of A-B pairs." Things like 'C' or '¬B' give no A without B, but they fail because they are not in the form of A-B pairs. Things like '¬A' or 'B' succeed when they are implicitly placed into the context of A-B pairs.
The point here is that if we sit down and think about what "No A without B" means in English, without assuming ahead of time that it means A?B, then we will recognize that it does not mean the same thing that A?B means. In some ways it does and in some ways it doesn't. My counterexample that began this whole tangent shows one of the ways that it doesn't.
If the idea here is, "It's not necessarily a good translation, but it's the best we have," then I would ask why it is better than the standard, "If A then B"?
I think A?B is better translated as, "If A then B."
¬(A?B) is suitably translated as, "A and not B."
The logical negation of A?B is different than an English negation, for the logical negation is more intuitively a negation of ¬A v B, which goes back to <this post>.
If one wants to make the negation translatable into English then "No A without B" is perhaps the best candidate, but it is not the best candidate apart from that single motive. Again, in propositional logic the negation of a conditional is never anything more than a counterexample, and this is the bug we are dealing with.
The solution you have arrived at is the idea that ¬(A?B) means, "A without B," and therefore (A?B) means "Not(A without B)." This misplaces the negations, acting as if the second negates the first when the opposite is true. What you are really saying is that ¬¬(A?B) means "Not(A without B)," and that (A?B) and ¬¬(A?B) are linguistically interchangeable.
What is really happening?
Then:
And then you assume that the '?' is applicable not only for logic, but also for English, thus:
This is almost identical to the problems in "Do (A implies B) and (A implies notB) contradict each other?" In both cases formal logical equivalence is being conflated with semantic equivalence.
The problem was isolated in <this post>. A?B and ¬(A?¬B) (or ¬A?B) are not the same sentence. A?B directly supports relations like causality, whereas the other two do not. Further, the only way to prove A?B from ¬(A?¬B) is via an indirect proof such as RAA, which is an equivalence and not a derivation. "If P then Q," and, "Not A-and-not-B" are two different claims, both in logic and in English.
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We can see this with an example.
A: I stop eating
B: I lose weight
The implication form is A?B ("If I stop eating, then I will lose weight"). This describes a relation between eating and weight. It means that to stop eating leads to losing weight, and that if one is not losing weight then they have not stopped eating (modus tollens).
The conjunction form is ¬(A?¬B) ("It is not the case that, it is true that I stop eating and it is false that I lose weight"). This says that A and ¬B cannot coexist. There is no relation posited between A and B.
The relation can be inferred from the conjunction, but it is not the same as the conjunction:
(4) follows from (1) and (2), but it is not equivalent to (1), despite the fact that the truth tables are the same. Put differently:
(One could also show this with RAA)
In your opinion arrows do not connote directionality? Do you think there is a reason logicians introduced the inference A?B over and above the conjunction ¬(A?¬B)?
Quoting Deleted user
Are you not equivocating between language speakers and non-language speakers? If only 2% of native speakers interpret ? as what we mean by 'dragon' in English then yes, it is a bad translation.
Quoting Deleted user
You think the English phrase, "No A without B," implies that we must be thinking about the entire domain of speech in terms of A-B pairs? This seems clearly incorrect. In English when we say, "No pizza without heartburn" we in fact order a salad ("C"), and this satisfies the condition just like Gregory's answer does.
Quoting Deleted user
You are again conflating the logic with the English. To think that the English entails whatever the logic entails is to beg the question and assume that the English perfectly maps the logic. That is what we are considering, not what we are assuming.
Regarding the modus tollens, the English does support it but, again, this is not the same as whether ¬A entails the truth of the conditional. These are not the same thing:
Specifically:
Does (4) hold? It is questionable, but if it doesn't then the translation limps, and if it does then this also holds: < C |= «No A without B» >, in which case the translation also limps since C does not semantically entail (A?B). Either way the translation limps.
Quoting Deleted user
But that's not what the English means. It is an arbitrary restriction of the English meaning. After all, if it's not being interpreted in favor of its literal meaning, then what is it being interpreted in favor of?
Part of the puzzle here is that in reality negations always obtain within a scope. For example, if C=salad, then C=¬A (not pizza). When we are within the same scope, C must always be either A or ¬A, and since C=¬A, C |= (A?B).
(Propositional logic seems to assume, prima facie, not only the commonsensical idea that C is neither A nor B, but also the deeply counterintuitive idea that C is neither ¬A nor ¬B. Usually if C is neither A nor B then it must be both ¬A and ¬B.)
Quoting Deleted user
What absurdities does it lead to?
A paper that shows how Medieval Aristotelian logic was in some ways more robust than current logic is Gyula Klima's, "Existence and Reference in Medieval Logic." He uses Russell's King of France example rather than conditionals.